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PROBLEMS OF RELATIVE GROWTH
HETEROCIfELY AND REGENERATION IN THE COMMON LOBSTER
i. Marine Lobster, Homarus (Gammarus) vulgaris, Normal Left-handed Specimen.
2. The Same Regenerating the Left 'Crusher' Claw after Autotomy.
3. Shed Skin of the Same at the Next Moult.
4. The Same after this Moult, showing Direct Regeneration of Left Crusher.
Note the slenderer propus and absolutely longer dactylus in the smaller (' nipper ') claw.
PROBLEMS
OF RELATIVE GROWTH
BY
JULIAN S. HUXLEY, M.A.
Honorary Lecturer in Experimental Zoology, King's College, London
WITH IO5 ILLUSTRATIONS
LINCOLN MAC VEAGH
THE DIAL PRESS
NEW YORK • MCMXXXII
PRINTED IN GREAT BRITAIN
TO
D'ARCY WENTWORTH THOMPSON
' The morphologist, when comparing one organism with
another, describes the differences between them point by point,
and " character " by " character." If he is from time to time
constrained to admit the existence of " correlation " between
characters (as a hundred years ago Cuvier first showed the
way), yet all the while he recognizes this fact of correlation
somewhat vaguely, as a phenomenon due to causes which,
except in rare instances, he cannot hope to trace ; and he falls
readily into the habit of thinking and talking of evolution as
though it had proceeded on the lines of his own descriptions,
point by point, and character by character. But if, on the
other hand, diverse and dissimilar fishes can be referred as a
whole to identical functions of very different co-ordinate
systems, this fact will of itself constitute a proof that a com-
prehensive " law of growth " has pervaded the whole structure
in its integrity, and that some more or less simple and recog-
nizable system of forces has been at work.' — D'Arcy Thompson
(Growth and Form, p. 727).
vn
PREFACE
IN this book I have attempted to give some account of
the chief results emerging from a study of the relative
growth of parts in animals which I have undertaken
during the last ten years. I have tried to correlate my own
findings and conclusions with those of other workers in the
same and related fields, but am well aware of the many gaps
that remain. However, it has not been my main intention
to produce an exhaustive survey of the subject, but rather
to set forth certain new facts and ideas and some of their
chief implications.
There are, I think, four chief points in the book which are
more or less new. One is the quantitative formulation of
heterogonic growth (Chapters I and II) ; a second is the dis-
covery of the widespread existence of growth-gradients, and
their quantitative analysis (Chapters III and IV) ; a third is the
recognition that growth of logarithmic spiral type as seen in
Molluscan shells, etc., operates with the same growth-
mechanisms (growth-centres and growth-gradients) as does
growth of ordinary type as seen in a Crustacean antenna or
a sheep's leg (Chapter V) ; and the fourth is the application
of these results to certain evolutionary problems, as set forth
in the final chapter.
I owe a great deal to previous work in this field : first and
foremost to D'Arcy Thompson's Growth and Form, but also
to the books and papers of Champy, Teissier, Schmalhausen ,
and others too numerous to mention.
I have to thank Professor L. T. Hogben and Dr. R. A.
Fisher, F.R.S., for reading the book in typescript and making
various useful suggestions ; and I owe a great deal to Pro-
fessor H. Levy for helping me with some of the mathematical
problems involved. My thanks are also due to Dr. C. F. A.
Pantin and Professor Selig Hecht, whose discussions with me
of various problems raised in this book have helped greatly
in clarifying my ideas. And especially I would like to thank
my pupils and co-workers, Mr. E. B. Ford, Mr. F. N. Rat-
cliff e, Miss M. Shaw (Mrs. White), Miss M. A. Tazelaar, Mr.
ix
x PROBLEMS OF RELATIVE GROWTH
S. F. Bush, Professor F. W. Kunkel, Mr. J. A. Robertson,
Miss I. Dean, Mr. A. S. Edwards and Mr. F. S. Callow, without
whose collaboration I should never have been able to collect
and analyse the data on which this treatment of the subject
is founded. Finally, I must not forget my secretary, Miss
P. Coombs, whose aid has been invaluable in preparing the
book for the press.
Many of the figures have been drawn for this book. As
regards the others, I would like to express my thanks for the
willingness of the authors and publishers concerned for allow-
ing me to reproduce them. Acknowledgements have been
made in the list of illustrations : the citations there made
refer to the literature list for fuller details.
JULIAN S. HUXLEY
King's College, London
December, 1931
ERRATUM
Page 84, Fig. 46, legend. For " dactylus ;
ischium " read " dactylus ; carpus ;
ischium "—problems of relative growth.
CONTENTS
PAGE
Preface ......... ix
List of Illustrations. ...... xiv
CHAPTER I
CONSTANT DIFFERENTIAL GROWTH-RATIOS
§ i. Introductory ........ i
§ 2. Constant Differential Growth-Ratios ... 4
§ 3. Examples of Constant Differential Growth-Ratios. 13
§ 4. Inconstancy of Form and Constancy of Form-change 38
CHAPTER II
THE COEFFICIENT OF CONSTANT GROWTH-
PARTITION ; AND SOME SPECIAL CASES
§ 1. The Heterogony of Deer Antlers .... 42
§ 2. The Coefficient of Constant Growth-partition . 49
§ 3. Heterogony in Holometabolous Insects 55
§ 4. Heterogony and Polymorphism in Neuter Social
Insects ......... 61
§ 5. Heterogony, Moulting, and Dimorphism ... 68
CHAPTER III
GROWTH-CENTRES AND GROWTH-GRADIENTS
§1
§2
§3
§4
§5
§6
§7
§8
Growth-gradients within Single Organs ... 79
Steepness of Growth-gradient within an Organ and
Growth-intensity of the Organ as a whole . . 83
Reversal of the Sign of the Growth-gradient in
Negative Heterogony ...... 87
The Form of Growth-gradients. . . . .90
Growth-gradients in Regions of the Body . . 92
Graded Growth-intensity in the Different Planes
of Space ......... 95
Gradients in Growth-rate of Epidermal Structures 100
Conclusion ......... 102
xi
xii PROBLEMS OF RELATIVE GROWTH
PAGE
CHAPTER IV
GROWTH-GRADIENTS AND THE GENERAL
DISTRIBUTION OF GROWTH-POTENTIAL IN
THE ANIMAL BODY
§ i. General Growth-gradients : D'Arcy Thompson's
Graphic Method . . . . . . .104
§2. General Growth-gradients: Quantitative Analysis iio
§ 3. The Two Phases of Growth . . . . .118
§ 4. Growth-changes Correlated with High Local Growth-
intensity ........ 120
§ 5. Some Cases of Teratological and Abnormal Growth 128
§ 6. The Law of Antero-posterior Development and its
Effect upon Growth ...... 132
§ 7. The Mathematical Formulation of Relative Growth
in Embryonic Life ....... 139
§ 8. Conclusion ......... 147
CHAPTER V
GROWTH-CENTRES AND GROWTH-GRADIENTS
IN ACCRETIONARY GROWTH
§ 1. The Accretionary Method of Growth . . . 149
§ 2. Logarithmic Spirals as the Result of Growth-
gradients . . . . . . . .151
§ 3. Growth-gradients and the Shells of Molluscs . 154
§ 4. Conclusion ......... 163
CHAPTER VI
HETEROGONY, GROWTH-GRADIENTS AND
PHYSIOLOGY
§ 1. Normal Proportions as Result of a Partition-equili-
brium ......... 165
§ 2. The Initial Determination and Physiological Basis
of Growth-gradients . . . . . .167
§ 3. Other Gradient Theories . . . . .170
§ 4. Heterogony and Hormones . . . . .176
§ 5. Heterochely and Relative Growth-rates. . . 1S9
§ 6. Specific Growth-intensities and their Interaction . 191
CONTENTS
Xlll
§ 7. The Influence of External Conditions
§ 8. Rhythmical Irregularities of Growth-ratio
page
197
203
§ 1.
§2.
§3-
§4-
§5-
§6.
§7-
§8.
CHAPTER VII
BEARINGS OF THE STUDY OF RELATIVE
GROWTH ON OTHER BRANCHES OF BIOLOGY
Heterogony and Taxonomy : Sub-species and Taxo-
nomic Forms ........ 204
Heterogony in Groups Higher than the Species . 212
Heterogony and Evolution . . . . .216
Heterogony and Comparative Physiology . . 224
Relative Growth and Genetics . . . .229
Relative Growth, Embryology, and Recapitulation 234
General Approach to the Problem of Qualitative
Form-change ........ 240
Conclusion ......... 243
Bibliography ........ 245
Addenda . . . . . . . ... 257
Index of Authors ....... 267
Subject Index ........ 270
Index of Organisms ....... 274
LIST OF ILLUSTRATIONS
FIG. PAGE
Heterochely AND REGENERATION IN THE LOBSTER
Frontispiece
(From Przibram, 1931)
i. Diagram illustrating some quantitative aspects of
heterogony ........ 5
2. Changes in absolute and percentage weight of the
large claw during growth in fiddler-crabs . 9
3. heterogony of the large claw in 4oi male fiddler-
CRABS ......... IO
(From Huxley, 1927k)
' 4. HETEROGONY OF STEM-WEIGHT AGAINST ROOT-WEIGHT IN
VARIOUS PLANTS . . . . . . .13
(From Pearsall, 1927)
5. HETEROGONY OF PETIOLE LENGTH AGAINST LAMINA DIA-
METER in Nasturtium leaves . . . . -14
(From Pearsall, 1927)
6. Changes in relative abdomen-breadth during the
growth of the shore-crab . . . . .15
7. heterogony of abdomen-breadth in 625 shore-crabs io
(After Huxley and Richards, 1931)
8. HETEROGONY OF THE CHELA IN THE PRAWN PALAEMON MAL-
COMSONI . . . . . . . . 17
g. heterogony of face relative to cranium in sheep-
dogs and baboons . . . . . . . 18
10. Baboon skulls of various ages to show change in pro-
portions ......... 19
(From Zuckerman, 1926)
11. HETEROGONY OF TAIL-LENGTH IN THE MOUSE PHENACOMYS 22
12. HETEROGONIC RELATION OF DORSAL AND VENTRAL EYE-
LOBES IN THE BAR-EYED MUTANTS OF DROSOPHILA . 22
(Modified from Hersh, 1928)
13. HETEROGONY OF INTER-OCULAR DISTANCE IN CRABS . 23
(After Teissier, 1931)
14. Change of proportions in shore-crab ... 24
(From unpublished drawings kindly supplied by Dr. G. Teissier)
15. HETEROGONY OF THORACIC GANGLION IN CRABS . . 25
(After Teissier, 1931)
16. HETEROGONY OF WATER-CONTENT IN TENEBRIO LARVAE 26
(After Teissier, 1931)
17. HETEROGONY OF NITROGEN-CONTENT IN TENEBRIO LARVAE 26
(After Teissier, 1931)
18. HETEROGONY OF PHOSPHORUS-CONTENT IN TENEBRIO
LARVAE ......... 28
(After Teissier, 1931)
xiv
LIST OF ILLUSTRATIONS xv
FIG. PAGE
ig. Heterogonic relation of heat of combustion to body-
weight in Tenebrio larvae ..... 28
(After Teissier, 1931)
20. Heterogony of water-content in wax-moth larvae
(Galleria) . . . . . . . -3i
(After Teissier, 1931)
21. Heterogony of male and female chelae in two species
of prawns (palaemon) ...... 33
(A, from Tazelaar, 1931)
22. Variation in the relative growth of the female
abdomen in fiddler-crabs ..... 35
(A, from Morgan, 1923 ; B, after Huxley, 1924)
23. Relative growth of male and female abdomen in
spider-crabs (inachus) ...... 36
(From Shaw, 1928)
24. ISOGONIC GROWTH OF PARTS IN THE FISH ORTHOPRISTIS . 37
(From Hecht, 1916)
25. Heterogony of antler-weight in adult Red Deer . 43
(From Huxley, 1931A)
26. Antler-weight against body-weight by age in Red
Deer ......... 44
(From Huxley, 1931A)
27. Negative heterogony of antler-weight in Roe Deer 46
(From Huxley, 1931A)
28. Diagram of antler-growth in Red and Roe Deer . 47
(From Huxley, 1931A.)
29. Body-weight and antler-weight against age, Red
Deer ......... 48
(From Huxley, 1931A)
30. Decrease of growth-coefficient during regenera-
tion, Sphodromantis ...... 50
(Modified from Przibram, 1917)
31. Regulation of size in grafted eyes of Amblystoma . 52
(From Twitty, 1930)
32. Interrupted heterogony in the male chela of spider-
crabs (Inachus) ....... 53
33. Heterogony of the fore-limb in the beetle Euchirus 56
(From Champy, 1924)
34. Heterogony of the ' tail ' appendage of the wting in
the butterfly papilio dardanus . . . -57
(From Champy, 1924)
35. Heterogony of the male mandible in three species
of stag-beetles (Lucanidae) ..... 58
(From Huxley, 1931c)
36. Change of proportions with increased size in neuter
ants (Pheidole) ....... 63
(From Wheeler, 1910)
37. Heterogony of head-size in neuter ants (Anomma
and Camponotus) ....... 64
38. Change of proportions with increased size in neuter
termites (Termopsis) ...... 66
(From Heath, 1927)
39. Dimorphism of female abdomen and male chela in
spider-crabs (Inachus) ...... 69
(From Shaw, 1928)
xvi PROBLEMS OF RELATIVE GROWTH
FIG. PAGE
40. Dimorphism and heterogony of the male forceps in
earwigs (forficula) . . . . . -71
(From Huxley, 1927s)
41. Diagram of the possible origin of forceps-dimorphism
in male earwigs . . ..... 73
(From Huxley, 1931c)
42. Differential effect of adverse conditions on
forceps-length and body-length in male earwigs 75
(From Huxley, 1927s)
43. Dimorphism and heterogony in the 3RD leg of the
mite Analges ........ 77
(From Jucci, 1924)
44. Difference in growth-coefficients of various regions
of the large claw of fiddler-crabs . . . 80
45. Growth-gradients within the male chela of crabs
(uca and maia) ....... 83
46. Changes in proportion of parts during growth of the
chela in prawns (palaemon) ..... 84
(From Dean, unpublished)
47. Growth-gradients in the antennae of Copepods . 86
48. Growth-gradients and evolutionary change in the
feet of Ungulates ...... 88
(From D'Arcy Thompson, 1917)
49. Reversed growth-gradient in the limbs of sheep . 89
(From Huxley, 1931B)
50. Change in form of growth-gradient with change of
growth-rate in the large claw of hermit-crabs
(eupagurus) . . . . . . . .91
(After Bush, 1930)
51. Growth-gradients in pereiopods and chelae of prawns
(Palaemon) ........ 92
(From Tazelaar, unpublished)
52. Growth-gradients in the abdomen of crabs (Pinno-
theres AND TELMESSUS) ...... 93
(From Huxley, 1931B)
53. Female pea-crabs (Pinnotheres) of various sizes,
showing changes in proportion of the abdomen
during growth ....... 94
(From Atkins, 1926)
54. Relative growth in the three planes of space in the
crusher and nipper claws of lobsters ... 97
55. Relative growth in length and breadth of the chela
in females, males and parasitized males of upogebia 98
56. Gradients in feather-growth in the fowl . . 101
57. Growth-gradients and evolutionary change in fish
(Diodon and Orthagoriscus) ..... 105
(From D'Arcy Thompson, 1917)
58. Cartesian transformations of the carapace of
various crabs . . . . . . .107
(From D'Arcy Thompson, 1917)
59. Reconstructions of evolutionary stages in the avian
pelvis, by the method of Cartesian transformation 108
(From D'Arcy Thompson, 1917)
LIST OF ILLUSTRATIONS xvii
FIG. PAGE
60. Reconstructions of evolutionary stages in the evo-
lution OF THE HORSE SKULL, BY THE METHOD OF CAR-
TESIAN TRANSFORMATION ; AND COMPARISON WITH
ACTUAL FOSSIL SKULLS . . . . . log
{From D'Arcy Thompson, 1917)
61. Growth-gradients along the body- axis of the hermit
crab eupagurus . . . . . . .112
{From Bush, 1930)
62. Growth-intensities of various parts in male and
female Stag-beetles (Lucanus) . . . .114
63. Growth-gradients of male and female stag-beetles 115
64. Growth-profile of metamorphosing herring . .116
{From Huxley, 1931B)
65. Effect of a region of high growth-intensity on
growth of neighbouring parts in spider-crabs
(Maia and Inachus) ...... 122
{A, from Huxley, 1927 a ; B, from Shaw, 1928)
66. Effect of a region of high growth-intensity on
growth of neighbouring parts in prawns (Palae-
mon) . . . . . . . . .124
{A, from Shaw, 1928 ; B, from Tazelaar, 1930)
67. Relative growth of parts anterior and posterior to
a region of high growth-intensity in prawns
(Palaemon) ........ 125
{From Tazelaar, 1930)
68. Effects of a regenerating limb upon the growth of
neighbouring limbs in Sphodromantis . . . 127
69. Graded growth-effects in two human monsters . 130
{A, from Nanagas, 1925 ; B, from Mead, 1930)
70. Change in relative weight of various organs of the
cat during fetal life . . . . . -134
{After Latimer and Aikman, 1931)
71. Positive heterogony of head-length in whalebone
WHALES ......... 137
72. Changes in relative weight of various organs of the
chick during embryonic life .... i43
{After Schmalhausen, 192713)
73. Growth-rates of various organs of the chick during
embryonic life ...... i44-i45
74. Diagram illustrating the co-operation of two
growth-ratios in determining the form of mol-
luscan shells . . . . . . 1 56
75. Origin of the spiral form of the shell in Limacina . 159
{From original by Dr. Lebour)
76. Diagram of the growth-gradients operating to pro-
duce THE TURBINATE SPIRAL SHELL OF MOLLUSCS . l6o
77. Heterogony of normal and regenerating claws in
portunus ........ 166
78. Asymmetry in the thoracic ganglia of the male fid-
dler-crab ........ 168
{From Ratcliffe, unpublished)
79. Chemical and metabolic gradients in Crustacea and
earthworms . . . . . . . .170
{From Perkins, 1929)
xviii PROBLEMS OF RELATIVE GROWTH
FIG. PAGE
So. Disproportionate effect of starvation on a hetero-
GONIC ORGAN (DORSAL CREST IN MALE TRITON) . . l8o
(From Champy, 1924)
81. Disproportion of limbs caused by precocious meta-
morphosis in the frog ...... 182
(Modified from Wells, Huxley and Wells, 'The Science of Life,'
London, 1931)
82. Differential effect of thyroidectomy on the growth
of various organs of the rat . . . 1 85
(From Hammett, 1929)
83. Effect of thyroidectomy on the growth of the hypo-
physis IN THE RAT . . . . . . I 87
(From Hammett, 1929)
84. Diagram illustrating Przibram's hypothesis of the
effect of differential growth-rates on chela-
reversal in heterochelous crustacea . . . igi
85. Growth of normal and grafted eyes in two species
of Amblystoma ...... 192-193
(From Twitty and Schwind, 1931)
86. Relative growth of eye in two species of Ambly-
stoma ......... 194
(From Twitty and Schwind, 1931)
87. Interaction of parts of the eye in Amblystoma . 196
88. Regulation of growth in eyes grafted on to indi-
viduals of different age ..... 198
(From Twitty, 1930)
89. Changes in relative weight of different parts of the
body caused by under-nourishment in rats . . 202
(From Jackson, 1925)
90. Comparison of absolute size and proportions of
modern and prehistoric scottish red deer . . 206
(After Ritchie, 1920)
91. The female and five different forms of male in the
stag-beetle cyclommatus ..... 209
(After Dudich, 1923)
92. Relative growth of mandible in the five male forms
of Cyclommatus . . . . . . .211
(From Huxley, 1931c)
93. Change of form with change of absolute size in the
mandible of male stag-beetles . . . .212
(From Grijfini, 191 2)
94. Heterogony in groups larger than the species : PHE-
NOMENON OF LAMEERE IN THE GENUS GOLOFA . 213
(After Champy, 1929)
95. Specific variations in the detail of a heterogonic
organ (cephalic horn) in goliath beetles . . 217
(From Champy, 1924)
96. Developmental changes of proportion in wild and
domestic sheep ....... 223
(After Hammond, 1928)
97. Graph showing relation of egg-weight to body-weight
in 432 species of birds ...... 226
(From Huxley, 1927c)
LIST OF ILLUSTRATIONS xix
FIG. PAGE
98. Diagram showing the effect of rate-genes on eye-
pigmentation in Gammarus ..... 228
(From Wells, Huxley and Wells, 'The Science of Life,' London,
1931)
99. Rate of eye-darkening in two genetic strains of
Gammarus ........ 230
(From Ford and Huxley, 192Q)
100. Effect of rate of general growth on rate of eye-
darkening IN A PURE STRAIN OF GAMMARUS . . 230
(From Ford and Huxley, 1920)
101. Multimodal frequency-distribution of body-build
index in man ........ 233
(From Davenport, 1923)
102. Diagram illustrating the effect of rate-genes upon
vestigial organs ....... 236
103. Effect of temperature on rate of eye-darkening in
a pure strain of gammarus ..... 238
(From Ford and Huxley, 1929)
104. Diagram illustrating the relation of mutations in
rate-genes to recapitulation and paedomorphosis 24o
PROBLEMS OF RELATIVE
GROWTH
CHAPTER I
CONSTANT DIFFERENTIAL GROWTH-RATIOS
§ i. Introductory
THE problem of differential growth is a fundamental
one for biology, since, as D'Arcy Thompson especially
has stressed (1917), all organic forms, save the simplest
such as the spherical or the amoeboid, are the result of dif-
ferential growth, — whether general growth which is quantita-
tively different in the three planes of space, or growth localized
at certain circumscribed spots. But the subject has received
little consideration. D'Arcy Thompson's own treatment,
though exhaustive on certain points (e.g. the logarithmic
spiral), profoundly original and important in others (e.g. his
use of Cartesian transformations to illuminate the evolution
of one form from another), and interesting throughout, is
admittedly incomplete. Certain large bodies of data, such as
those included in Donaldson's The Rat (1924) and in various
treatises on physical anthropology, e.g. R. Martin (1928 ) exist
on differential growth in mammals, but have so far not been
analysed save by the use of purely empirical formulae ; Champy
(1924) has written a very stimulating book on differential
growth of such extreme type as to warrant the term ' dys-
harmonic ', and has later given further examples (1929) ;
Przibram has recently (1930) collated some of his interesting
results and ideas. But, apart from this, little that is con-
nected or general has been written on the subject ; and even
the individual papers dealing with the topic are few and on
the whole disconnected.
Since 1920 I have been studying certain phases of the
problem : the purpose of the present review is to bring to-
gether the various aspects which have presented themselves,
2 PROBLEMS OF RELATIVE GROWTH
to demonstrate the existence of certain broad empirical laws
which appear to govern most cases of differential growth so
far studied, to discuss their bearing on other branches of
biology, and to point the way to further attack on the subject
by those trained in other methods.
The first step, it appeared to me, was to study a number
of clear-cut cases of differential growth and to see whether
they were capable of quantitative expression. My own mathe-
matics are regrettably deficient, but I was able (see Section 2)
to obtain a simple formula which appears to be at any rate
a first approximation to a general law for differential growth.
Among many morphologists and systematists there appears
still to linger a distrust of the application of even such element-
ary mathematics to biological problems. The usual criticism
is that the formulae arrived at may have a certain convenience,
but can tell us nothing new, and nothing worth knowing of
the biology of the phenomenon. This appears to me to be
very ill-founded. In the first place, to have a quantitative
expression in place of a vague idea of a general tendency is
not merely a mild convenience. It may even be a very great
convenience, and it may even be indispensable in making
certain systematic and biological deductions. But further, it
may suggest important ideas as to the underlying processes
involved ; and this is precisely what the quantitative analysis
of relative growth is doing. As will be seen in this and the
subsequent chapters, there are certain hypotheses which square
with the formula, others which do not : without the quanti-
tative expression, we should be largely theorizing in the air.
I would not trouble to spend my time on this point if it had
not been urged on several occasions in my hearing ; other-
wise, one would expect that the interaction of quantitative
theory with observation and experiment devoted to testing
the theory, so fruitful not only in other sciences but in
genetics within the field of biology, would automatically be
welcomed.
Furthermore, the establishment of one quantitative rule
leads on to the discovery of others. Chapters I and II will
be devoted to showing that, when we consider the growth of
whole organs relative to the rest of the body, the results can
be understood if we postulate that the ratio between the
intensity (or relative rate) of growth of the organ and that
of the body remains constant over long periods of the animal's
life. To borrow a term from another branch of science, there
INTRODUCTORY 3
is a constant partition-coefficient of growth-intensity between
organ and body. It was next found that in many organs,
especially those growing at markedly different rates from the
body as a whole, growth-intensity was not distributed uni-
formly, but in a more or less regular pattern. This led on
to the notion, already suggested on different grounds by
D'Arcy Thompson, that the growth-intensity of the body as
a whole (or, if you prefer it, the relative growth-rates of its
various parts) is distributed according to an orderly system
of ' growth-gradients '. These conclusions will be discussed
in Chapters III to V.
The physiological mechanism underlying these general rules
still remains very obscure, in the absence of experiment
specifically directed to the point : but there are some inter-
esting hints and possibilities, and these will be discussed in
Chapter VI.
Finally, the facts derived from the study of relative growth
have a number of important bearings upon other branches of
biology ; and the concluding chapter will be devoted to these.
I hope to convince the systematist that by a knowledge of
the laws of relative growth we are put in possession of new
criteria bearing on the validity of species, sub-species, and
' forms ' ; the nature of certain dwarf forms ; and the import-
ance (or the reverse) of size-differences in general for system-
atics. In regard to that special branch of systematics usually
called physical anthropology, it will be found that these laws
have a bearing on the important question as to whether true
evolutionary change has taken place in civilized populations
during historical time. As regards evolution, it will be found
that the subject throws light upon the question of adaptation,
on the general theory of orthogenesis, and on the selection
problem. Furthermore, the existence of growth-gradients, as
D'Arcy Thompson has already pointed out, makes it much
easier for us to understand how certain types of evolutionary
transformation can have been brought about, since a single
genetic change affecting a growth-gradient will automatically
express itself in a changed relation in the size of a large
number of organs or regions.
Then comparative physiologists will find it necessary to
know precisely how to discount the effects of differences in
total absolute size when they wish to estimate the compara-
tive development of an organ in a series of related species
or groups ; and will further find interesting hints as to the
4 PROBLEMS OF RELATIVE GROWTH
nature of factors which tend to limit the size of an organ
at high absolute sizes.
Nor can genetics be left out. A constant partition of
growth-intensity between different regions implies constant
differences in their rates of growth. Thus any genes control-
ling relative size of parts will have to exert their action by
influencing the rates of processes, and so fall into line with
the numerous other rate-factors whose importance has been
summarized by Goldschmidt (1927) and by Ford and Huxley
(1929). The fact, however, that the ratios between growth-
rates, and not their absolute values, are the determining
factors introduces certain complications, whose discussion will
be found to have an interesting bearing upon the analysis of
other genetic ' characters '.
Finally, the ancient problem of embryological recapitulation
will be found to be illuminated from a new angle ; and many
undoubted cases of recapitulation will be found to owe their
origin not to any mysterious phyletic law, but to embryological
convenience, adjusting evolutionary changes in the size of an
organ to the general rules of relative growth during individual
development.
This brief introductory sketch will, I hope, have shown
some of the chief points of interest in the study of relative
growth. We must now come to grips with the subject,
and for the reasons above stated propose to do so by con-
sidering what at first sight seems a rather arid point — the
quantitative expression of the relation between the body as
a whole and an organ whose proportionate size changes during
life.
§ 2. Constant Differential Growth-ratios
Champy (1. c.) and others have pointed out that certain
organs increase in relative size with the absolute size of the
body which bears them ; but so far as I am aware, I (Huxley,
1924B) was the first to demonstrate the simple and significant
relation between the magnitudes of the two variables. In
typical cases, if x be the magnitude of the animal (as measured
by some standard linear measurement, or by its weight minus
the weight of the organ) and y be the magnitude of the dif-
ferentially-growing organ, then the relation between them is
y = oxk, where b and k are constants.1 The constant b is
1 This can also be written log y = log b + k log x, which means that
any magnitudes obeying this formula will fall along straight lines if
plotted on a double logarithmic grid.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 5
here of no particular biological significance, since it merely
denotes the value of y when x = 1 — i.e. the fraction of x
which y occupies when x equals unity. We may call it the
3000
I
^2000
I
•is
"a
1000
500
100
k=20
1 ' ■ ■ 1
b =
- OO 003
h =
f® /•/ x I0'5
® /Ox/0"5
/
k -27 <
®09*I0'S d
O0-002
b - /0~5x/-7+\
1
1
1
1
1
1
f
t . \H+tr
- O 0-001
b - /0 x i/o 9
/
t
/
■ O00005
/ 05 + S
/k = l3 (8/0 )x/0~5= h
S [(DOS )
0
<
o-oooi s
eJOdbOOl
J 000 5000 10,000 15,000
size of body
-3000
2000
1000
Fig. 1. — Diagram to show the quantitative effect of varying the constants
in the simple heterogony formula, y = bxk, assuming that the origin of growth
in x and y begins at the same time.
The dotted line gives the growth of the organ (y) when k = 2-0 and b = -ooooi. The points to
the left show values of y for different values of b when the rest-of-body is of size 1000. Those to the
right show the effects, at body-size io,ooo, of varying both k and b.
6 PROBLEMS OF RELATIVE GROWTH
fractional coefficient. But the value of k has an important
meaning.
It implies that, for the range over which the formula holds
the ratio of the relative growth-rate of the organ to the relative
growth-rate of the body remains constant, the ratio itself
being denoted by the value of k. By relative growth-rate is
meant the rate of growth per unit weight, i.e. the actual abso-
lute growth-rate at any instant divided by the actual size at
that instant.
This is at once seen by plotting the logarithm of y against
the logarithm of x. In unit time the increase in the logarithm
of y is k times the increase in the logarithm of x, which may
be written :
d , ■, d ,
g.logy-Jglog*
dy , ,dx ,
ily - kTtlx
This formula, on which I have had the advantage of con-
sulting Professor Levy, of the Imperial College of Science,
can be deduced on the basis of very simple assumptions about
growth in general. One essential fact about growth is that
it is a process of self-multiplication of living substance — i.e.
that the rate of growth of an organism growing equally in
all its parts is at any moment proportional to the size of the
organism. A second fundamental fact about growth is that
the rate of self -multiplication slows down with increasing age
(size) ; a third is that it is much affected by the external
environment, e.g. by temperature and nutrition. The two
latter considerations affect all parts of the body equally, so
that we may suppose that the growth-rate of any particular
organ is proportional simultaneously (a) to a specific constant
characteristic of the organ in question, (b) to the size of the
organ at any instant, and (c) to a general factor dependent
on age and environment which is the same for all parts of
the body.
If y stand for the size of the organ, and x for that of the
rest of the body, we shall then have
— = oaG and -- = #yG,
dt at
where a and /? are the specific constants for the rest of the
body and for the organ in question, and G measures the
CONSTANT DIFFERENTIAL GROWTH-RATIOS 7
general conditions of growth as affected by age and environ-
ment,1 then — = — •
ax ax
Thus log y = - log x + log b, where b is a constant : i.e.
y = bxp/a.
And fi/a, which can also be written k, is a constant, and is
also the ratio of the specific components of the growth-rates
of y and x respectively.2
I am, of course, aware that the existence of growth-cycles
and other facts make it impossible to suppose that the ex-
pression for change of growth can be so simple as here set
forth. We must suppose that each cycle may have its own
general and specific components of the growth-rate — i.e. that
a, /5 and y may change comparatively abruptly during the
life-cycle, and also it is quite possible that other inherent
alterations, such as the gradual increase of viscosity of proto-
plasm with age (Ruzicka, 1921), will cause gradual and pro-
gressive diminution of the specific constants which would
account for the various distortions of the S-shaped curve of
growth from the form expected on the simplest assumptions.
But I am convinced that some such general method of en-
visaging growth is sound ; and it is interesting to find our
empirical formula for constant differential growth-ratios
deducible from it. (See also Schmalhausen, 1927B, 1930.)
Exactly the same formula would apply to two sums of
money put out at different rates of compound interest, pro-
vided that they were not accumulating discontinuously by
quarterly or annual interest payments, as in financial fact,
but continuously, as in the Compound Interest Law, and as
in biological growth, k would here denote the ratio of the
1 One might expect, from certain experimental data, that the factor
G would be a simple function of the defect of the size of the organism
at any given time from its final size ; but this would not interfere
with the validity of our more general formula.
2 It may well be that the ' general factor ' is not capable of such
a simple formulation. But provided that such a general factor does
exist — i.e. that the growth both of organ and of rest-of-body is related
to some general law of growth affecting the organism as a whole, the
deduction of constant differential growth-ratios remains valid. And
that such a relation does exist is shown by the work of Przibram,
Harrison and others discussed in Chapter VI.
8 PROBLEMS OF RELATIVE GROWTH
two rates of interest. (In our biological parallels, we know
nothing of the actual rates of growth, for since the organ and
the body have both existed for the same length of time when
we measure them, the time-factor cancels out, in point of fact.)1
The actual rates, unlike those for the two sums of money, will
obviously be altering continuously; they will be high in youth,
low in age ; increased by high temperature, decreased by low ;
and so forth. What concerns us is that if our formula holds,
the ratio of the relative rates of growth remain constant.
In such cases, therefore, we have a constant differential
growth-ratio , denoted by the value of k. If we prefer to con-
centrate upon the growth of the organ relative to the growth
of the body considered as a standard, then we may speak of
k as denoting the growth-coefficient of the organ. An organ
which is thus growing at a different rate from the body as a
whole may be called heterogonic, to use the convenient term
coined by Pezard (1918). If it is growing at the same rate
as the body it must be styled isogonic ; as will be apparent,
isogony is merely a special case of heterogony, as the circle
is a special case of the ellipse.
It is clear that comparatively small variations in the value
of k will have large results provided that growth continues
over a considerable range of size. An attempt to show this
graphically has been made in Fig. 1.
The best worked-out example of this law so far concerns
the large chela of male fiddler-crabs, Uca pugnax (Huxley,
1927A). This obeys the law of constant growth-ratio from crabs
of only about 60 milligrams total weight to the largest found,
weighing sixty times as much ; the value of k, however, changes
quite abruptly at about i-i g. total weight, a point which
probably denotes the onset of sexual maturity, decreasing here
to less than 80 per cent, of its value for the earlier growth-
phase. (It is a noteworthy and unexpected fact that the
growth-coefficient of this secondary sexual character should
be reduced instead of increased when the gonad begins to
function.) See Figs. 2 and 3.
In our examples of the fiddler-crab, the weights of chela and
rest-of-body behave, over the earlier and longer growth-phase,
like £2 and £100 put out at 8 per cent, and 5 per cent, (continuous)
compound interest respectively 2, and a calculation on this basis
1 For cases where an organ is laid down considerably later than
the body as a whole, see Chapter IV.
2 Strictly speaking, of course, 8-i and 5 per cent, (see p. 10).
CONSTANT DIFFERENTIAL GROWTH-RATIOS 9
will reproduce the actual figures for weight.1 But we can be
perfectly sure that the actual growth-rate of the crab and of
its claw slows off with age, that it differs in summer and
winter, and is further subjected to all kinds of irregular fluc-
tuations due to temperature, food and other factors. The
actual rates may be as 8,000 : 5,000 in early life, as 160 : 100
later, as 4 : 2-5 in maturity, and as 0-08 : 0-05 in extreme old
age ; yet so long as the ratio 8 : 5 is preserved, claw-size will
always be the same function of body-size — a body of given
0 100 ZOO 300 400 600 BOO 1000 IZSO H00 1750 Z000 ZZSO
weight of rest of body.mg
Fig. 2. — Increase of absolute and relative chela-weight in the large chela of
the male fiddler-crab, Uca pugnax.
(Constructed from the data of Huxley, 1927A.)
weight will have attached to it a claw whose weight would
be the same whether the body had taken three weeks or three
years to reach its present size. 2 The differential growth-ratio re-
mains the biologically and morphogenetically important factor.
The expression^ = bxk can be written log y = log b + k log x.
1 y0 = 2, x0 = 100. At time t, y, = y0eom, xt = A'oe005(. After ten
years, y10 = 2eos = ^4-45, and x10 = iooe0'5' = ^164-92. After twenty
years, y20 = ze16 = ^9-9i. and x20 = iooe10 = £271-9, and so on.
Double logarithmic plotting of these figures gives a straight line.
2 In all probability this is only true as an approximation. It is a
priori unlikely that there is no differential effect of environmental
agencies on the growth-rates of body and chela respectively.
10
PROBLEMS OF RELATIVE GROWTH
In other words, if the logarithms of the magnitudes are plotted,
we should expect a straight line, from the slope of which the
value of k can be read off ; (if a be the angle the line makes
with the x axis, then tan a = k). Fig. 3 shows the excellent
1.5
2.0
2.5 3.0
I
3.0
1
1 1 Q
rT
2.0
Oj
0
1.0
yO
J> Upugnflxb"
M 1
1
3.5
I
H3.0
2.0
1P
17 2.0
0.5
3P 3.4
Fig. 3. — Increase of the logarithm of absolute chela-weight with the logarithm
of body-weight in male fiddler-crabs.
approximation of the actual points to a straight line when so
plotted. In this particular case, the constants are as follows
— first phase (to total weight i-i g. or just over ; rest-of-body
weight about 075 g.) : b = 0-0073, k = 1-62 ; second phase
(from this point to maximum size, in this sample maximum
CONSTANT DIFFERENTIAL GROWTH-RATIOS u
total weight rather over 3-5 g.) : b = 0-083, k = I#255- As
purely graphic methods, especially with logarithmic plotting,
are not sufficient to establish the accuracy of an empirical
formula of this sort (see, e.g., Gray, 1929), I have calculated
the values to be expected from the formula. As will be seen
from Table I, the deviations from expectation are slight —
only in four cases over 5 per cent., and these all in the first
phase, where errors in weighing are liable to be relatively
greater ; the mean deviation for the second phase is only
+ 0-19 per cent., for the second phase it is + 0-35 per cent.,
and would be smaller but for the large deviation of the last
class, which consists of only a few individuals. Further, there
is no trend of the deviations from predominantly positive to
predominantly negative or vice versa. We may thus take
the formula as a rather surprisingly close approximation to
reality. It is possible that the delimitation of the beginning
of the second phase after the 14th instead of after the
15th class would have improved matters ; and also that
small alterations in the values of k would have given an
even better fit,1 but I am only concerned to show that the
data conform to this type of mathematical expression, not
to obtain accuracy in an extra decimal place in the formula
itself.
We are accordingly justified in saying that the large chela
of the male Uca grows in close approximation to the formula
of constant differential growth-ratio, namely : y — bxh.
In passing, it is worth noting that the logarithmic method
of plotting brings into true relief an important point which
is entirely obscured by the usual method of plotting on the
absolute scale — namely the fact that growth is concerned
essentially with the multiplication of living substance. On
the logarithmic scale, equal spaces on the graph denote equal
amounts of multiplication, whereas on the ordinary absolute
scale they denote equal additions. From the point of view
of growth, the increase of weight of our fiddler-crabs from
5 mg. to 25 mg. is equivalent to that from 1 g. to 5 g. ; but
on the absolute scale the former interval cannot even be repre-
sented on the same graph as the latter. Thus when I speak
of a fraction of the growth-period, I shall invariably be think-
ing in terms of multiplicative growth, in which an «-fold
1 The last two columns of the table give the expectation if 0-0074
be substituted for 0-0073 as the value of b, and show that this gives
a greater deviation, but one of opposite sign.
12
PROBLEMS OF RELATIVE GROWTH
TABLE I
Uca pugnax (401 Specimens) Growth-ratio of Large Chela {y)
and Rest of Body (x)
(a) First phase: to total weight (x + y) = i-i g. y = 0-0073 *162 (mg-)'
# = mean
weight cf
y = mean
Per cent.
y calculated
Per cent.
rest of
weight of
y
deviation
on formula
deviation
body after
large chela
calculated
actual from
y =
actual from
removal of
large chela
(actual)
calculated
o-oo74sr1'62
calculated.
mg.
mg.
mg.
mg.
I
57'6
5'3
5-16
+ 2-7
5-24
+ I-I
2
80-3
9-0
8-89
+ 1-2
9-02
— 0-2
3
109-2
13-7
14-59
- 6-i
14-79
- 7-4
4
156-1
25-1
25-88
- 3-o
26-24
- 4-0
5
199-7
38-3
38-90
- i-5
39-45
— 2-9
6
238-3
52-5
51-76
+ 1-4
52-48
4- 0-4
7
270-0
59-o
63-53
- 7-i
64-42
-8-4
8
300-2
78-1
75-34
+ 3-7
76-38
+ 2-3
9
355-2
104-5
98-63
+ 5-9
ioo-o
+ 4-5
10
420-1
i35-o
129-4
+ 4-3
131-2
4" 2-9
11
470-1
164-9
155-2
+ 6-2
157-4
+ 4-8
12
535-7
195-6
191-9
+ i-9
194-5
4- o-6
13
617-9
243-0
242-7
+ o-i
246-0
— 1-2
14
68o-6
271-6
283-8
- 4-3
287-7
-5-6
15
743-3
319-2
327-3
- 2-5
331-9
-3-8
Algebraic sum of deviations 4- 2-9
Mean deviation 4-0-19
— 16-9
- i-i3
(6)
Second phase : from total weight 1-2 g.
onwards.
y = 0.
y calculated
y
Per cent.
on formula
X
y
calculated
deviation
y =
0-084 S1'256'
Per cent,
deviation
16
872-4
417-6
406-8
4- 2-6
411-7
4- i-4
17
983-1
460-8
472-4
- 2-5
478-0
-3-6
18
1,079-9
537-o
531-9
+ I-I
538-3
— 0-2
19
^^SS
593-8
585-6
4- i-4
592-7
4- 0-2
20
1, 211*7
616-8
628-7
- 1-9
636-2
- 3'i
21
1,291-3
670-0
665-6
+ 0-7
673-6
- 0-5
22
1,363-2
699-3
720-6
- 3-o
729-3
- 4-i
23
1,449-1
777-8
769-1
4- 1-7
778-4
— o-i
24
1,807-9
1,009-1
1,015-0
- o-6
1,028-0
- i-8
25
2,235-0
1,380-0
1,327-0
+ 4-°
1,344'°
4- 2-6
Algebrai<
; sum of deviations
+ 3-5
- 9-2
Mean deviation
+ o-35
— 0-92
In Huxley, 1927A (p. 152), this was stated as 1-33 owing to an error.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 13
increase from one absolute size is regarded as equivalent to
an «-fold increase from another absolute size.
The same total increase could be subdivided into fractions
of equal absolute size ; but this method of subdivision in
terms of additive growth has not the same biological value,
and will not be adopted. Thus for an increase from 1 g. to
256 g., equal fractions of the growth-period are best repre-
sented by the equal multiplicative increases from 1 to 4, 4 to 16,
16 to 64 and 64 to 256 g. ; and not by the equal additive
increases to 64, 128, 192 and 256 g.
§ 3. Examples of Constant Differential Growth-ratios
The expression y = bxu I shall refer to as the simple hetero-
gony formula. This formula, or an approximation to it, has
been found to hold for a number of other organs, e.g. the
Fig. 4. — Increase of the logarithm of stem-weight against the logarithm of
root-weight in various plants.
14
PROBLEMS OF RELATIVE GROWTH
abdomen of some female crabs (Shaw, 1928 ; Sasaki, 1928) ;
the chelae of many male and some female Decapoda (Huxley,
1927 ; Shaw, 1928 ; Tazelaar, 1930 ; Tucker, 1930 *) ; other
appendages of various Crustacea ; the trunk of Planarians as
Fig. 5. — Increase of the logarithm of petiole-length against the logarithm
of lamina diameter in nasturtium leaves, Tropaeolum.
against the head (Abeloos, 1928) ; the face as against the
cranium of dog and baboon (Huxley, 1927, analysing Becher,
1923 ; Huxley, unpublished, analysing Zuckerman, 1926) ;
the shoot as against the root of certain plants (Pearsall, 1927) ;
the size (facet-number) of the two lobes of the eye in the
1 Tucker states that his data indicate a linear relation between male
chela length and breadth, and carapace length. However, his graphs
and his percentage measurements indicate that this does not give an
accurate fit, whereas logarithmic plotting (Fig. 55) gives an excellent
approximation to a combination of two straight-line curves, as in
male Uca.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 15
bar-eye mutant of Drosophila melanogaster (Hersh, 1928) ;
the linear dimensions of certain Molluscs (Nomura, 1928 ;
30" "To
/o
40-
30-
20-
10
-— — ---»t-.
10
20
30
40 50 60
mm
Fig. 6. — Increase in relative width of abdomen with increase of carapace
length in the shore-crab, Carcinus maenas : above, female ; below, male.
The ordinates represent °_?5 — \ O/ • f he abscissae represent carapace length in mm.
carapace length /<J '
Nomura and Sasaki, 1928) ; the tail of the mouse Phenacomys
(Taylor, 1915) ; the dimensions of the casques of Hornbills
(Banks, unpublished) ; the weights of various organs of the
i6
PROBLEMS OF RELATIVE GROWTH
rat (analysis of the data of Donaldson, Hatai, Jackson, etc.,
— see Donaldson, 1. c.) ; the length of the head of whalebone
whales during post-natal life (Mackintosh and Wheeler, 1929) ;
the lamina diameter and petiole length in Tropaeolum leaves
6 8 10 20
carapace length, mm.
30 40 50
Fig. 7.
-Increase of width of abdomen with increase of carapace length in
the shore crab, Carcinus maenas : logarithmic plotting.
+■ , unsexables ; 0, females ; H males. The growth coefficient for unsexables and young females is
1-26, for older females 1-42, that for young males 107, for older males 0-94.
(Pearsall, 1927), and even for the amounts of various chemical
substances in the growing larvae of the wax-moth Galleria
and the meal-worm beetle Tenebrio (Teissier, 1929, 1931), as
well as for organs which, physiologically speaking, represent
special cases, as the antlers of deer which are shed every year
CONSTANT DIFFERENTIAL GROWTH-RATIOS 17
PaJaemon
malcomsoni
2 6
2 4
2 2
2 0
(Huxley, 1926, 1927, 193 1), and the horns, mandibles, etc.,
of various holometabolous insects (Huxley, 1927 — see Chapter
II). It is probable that any other changes of proportion which
have not yet been analysed from this point of view will turn
out to obey the same law, e.g.
the progressive increase of rela-
tive tail-length in the salamander
Eurycea (Wilder, 1924), or that
of wing-rudiments in dragonflies
(Balfour-Browne, 1909). See also
p. 258 (Daphnia), p. 263 (rat).
Most of the organs thus far
cited are positively heterogonic,
increasing in relative size with
growth ; others, however, de-
crease in relative size with
increase in absolute size, and
are accordingly negatively hetero-
gonic. Such of these cases as
have been analysed also appear
to obey the rule of constant
differential growth-ratios : e.g.
nucleus in oocytes of Hydractina
(Teissier, 1927) ; brain in vari-
ous mammals (Lapicque, 1907 ;
Dubois, 1914, 1918) ; the number
of nerve-fibres and /or neurones
in mammals (Lapicque and
Giroud, 1923 ; Dubois, 1918) ;
heart in many vertebrates (Klatt,
1919 ; Clark, 1927, for references) ; limbs in post-natal sheep
(Hammond, 1927, 1929) ; certain limbs in Hermit-crabs (Bush,
I93°) '> pereiopods in the racing-crab Ocypoda (Cott, 1929) ;
legs in Orthoptera (Przibram, 1930) ; Gammarus eyes (p. 260).
Some of the facts are graphically illustrated in Figs. 4, 5
(plant organs) ; 6, 7, 22, 23 (crab abdomen) ; 8, 21, 32, 55,
77 (Crustacean chelae) ; 9, 10 (mammalian cranium) ; 11
(mouse tail-length) ; 12 (facet-number, Drosophila eye) ; 13-15
(various organs of crabs) ; 16-20 (chemical substances in insect-
larvae) ; 71 (head-length, whales) ; 25, 27 (deer antlers) ;
33-35, 40, 91, 92 (organs of holometabolous insects).
We may give some tables and figures in support of these
statements.
log body length, mm.
Fig. 8. — Relative growth of the
chela in the prawn, Palaemon mal-
comsoni : logarithmic plotting.
(From the data of Kemp, and Henderson and
Mathai.)
i8
PROBLEMS OF RELATIVE GROWTH
Sheep Dog (Fig. 9)
Data from Becher (1923)
Analysed in Huxley, 1927
x
42-0
05-3
74-5
85-5
99-3
II2-6
I20-O
y
22 -o
48-3
58-0
73-5
89-1
102-0
II2-0
x
y
k
30-
cranial region (mm.).
facial region (mm.).
1-5 (except for highest values of x).
160
— 1 \ 1 —
-
130
-
I/K
100
•
r
-
■
5?
«5!
■
/ 1
/ 1
-
1 ™
/ X
7 '
/ 1
/ 1
/ 1
1
1
(6)
■
20
1
— 1 1 1
1
40 60 80 100
cranium length, mm.
120
Fig. 9. — Growth of the facial region relative to the cranial region in the skulls
of sheep-dogs (x) and baboons (©).
k for sheep-dog about 1-49 ; for baboon, points 2-5, about 4-25.
[Constructed from the data of Becher, 1923, and Zuckerman, 1926.)
CONSTANT DIFFERENTIAL GROWTH-RATIOS 19
Values of Cranium-length and Face-length in the Baboon Papio
porcarius at Different Sizes (from Zuckerman, 1926) (Figs. 9, 10)
No. of Cases
1
4
7
3
6
4
The growth-coefficient of face-length on cranium-length for
Classes 2-5 is about 4-25, a very high figure. The curve
shows irregularities at both ends.
Mean Face-
Mean Cranium-
length (Naso
length
prosthion)
mm.
mm.
• 78-5
31-0
100-25
64-6
108-9
94-8
• "47
131-0
. 118-25
140-8
122-0
M4-25
Fig. 10. — Baboon skulls of various sizes, to show the increase in relative size
of facial region with absolute size of skull.
1, new-born; 2, juvenile (with milk dentition); 3, adult female; 4, adult male.
20
PROBLEMS OF RELATIVE GROWTH
TABLE Ia
Abdomen-breadth against Carapace-length in Carcinus
MAENAS FROM PLYMOUTH (FigS. 6, 7)
(Data of Huxley and Richards, 1931)
Mean
Mean
V
car.-l.
abd.-br.
(mm.)
(mm.)
(a) Unsexable (74 specimens)
13
3-09
0-578
22
3-80
0-680
20
4-22
0-823
11
476
0979
8
5-19
1-019
(b) Fern
ales (281 specimens)
12
5-56
1-16
16
6-52
i-45
14
7-41
1-67
12
9-32
2-30
15
10-37
2-80
16
n-35
3-n
17
12-33
3-37
23
13-29
3-81
19
14-35
4-06
12
I5-3I
429
19
16-35
4-82
15
17-36
5-15
15
18-16
5-48
7
19-34
5-93
6
20-33
6-72
10
21-51
6-82
12
22-40
7-54
8
23-34
7-76
6
24-33
8-33
5
25-52
9-16
10
26-45
9-59
6
27-65
9-79
2
28-45
10-40
3
29-30
1 1 03
4
3o-55
11-62
4
31-49
12-46
3
32-37
12-75
5
33-32
12-70
4
34-33
I3-56
1
43-5o
17-50
2
45-3o
19-20
1
46-30
19-90
1
50-20
23-70
CONSTANT DIFFERENTIAL GROWTH-RATIOS 21
Values of Constant Differential Growth-ratios for Shoot-
weight against Root-weight in Various Plants (from Pear-
sail, 1927) (Fig. 4)
Plant
Value of k
Daucus carota (carrot)
°'55
Brassica rapa (turnip)
0-65
Gossypium roseum (cotton)
090
Impatiens sp. ....
i-oo
Pisum sativum (pea). .
0-90-1-15 (3 expts.).
,, ,, (etiolated) .
1-75-2-65
Triticum vulgare
1-05
Hordeum distichum (low N)
1-20
(highN)
i-55
Linum usitatissimum
1-30
Values of Cell-diameter and Nuclear Diameter in the Oocytes
of Hydractinia Echinata, in /i. (From Teissier, 1927)
k = 0-69 ; b — 1-5
x = oocyte diameter
y = nuclear
Deviations from cal-
culated values
6-8
5-6
o-o
io-o
7-3
o-o
13-6
9-0
— o-i
17-5
1 1*4
+o-i
22-2
12-5
— 0-2
25-3
14-4
+ 0-5
34'°
16-9
— 0-2
x = oocyte diameter
y = nuclear ,,
Deviations from cal-
culated values .
43-o
22-5
+ 2-4
53-5
24-8
+ i-5
70-0
29-6
+ i-5
•88-o
3o-3
-2-6
n8-o
36-5
-3-8
136-0
42-7
-2-5
168-0
52-0
-f-O'2
Mean Values of Pre-ocular Length and Total Length in Planaria
gonocephala, calculated from Abeloos (1928)
k — 0-63 approx.
Total length . 1-5 3-0 50 18-0 mm.
Pre-ocular length . 0-231 0-43 0-50 1-125 mm.
Mean Values for Facet-Number in the Two Lobes of Bar-eyed
Drosophila at Different Temperatures (from Hersh, 1928) (Fig. 12)
Temperature
32-0°
29-5°
27-5°
25-5°
21-5°
18-0°
15-0°
No. facets
dorsal lobe
No. facets
ventral lobe
17-63
12-73
22-15
18-86
23-30
19-24
28-63
19-11
43-35
36-24
62-08
57-62
66-04
67-28
I Homozy-
rgous bar
J ??
No. facets
dorsal lobe
No. facets
ventral lobe
81-22
41-93
152-15
9020
150-29
103-66
193-54
114-29
196-21
151-51
222-45
I59-OI
256-12
185-89
\ Heterozy-
1 gous (Bar
[ X normal)
) 99
In both cases k for ventral lobe on dorsal lobe is about 1-5,
but the value of b is considerably higher for the homozygotes.
Fig. i i . — Tail-length against total length
during growth in the mouse, Phenacomys
longicaudus ; logarithmic plotting.
k = about i -41.
(Recalculated from the data of Taylor, 1915, p. 129.)
100 130 160 ZOO
total length, mm.
15
20
30 40 50 60 70 8090100
dorsal facet number
150 ZOO
300
Fig. 12. — Relation of facet-number in dorsal and ventral lobes of mutant
female fruit-flies of the bar-eye series ; logarithmic plotting. With decreasing
temperature, the total number of facets in the eye increases ; but the number
in the ventral lobe of the eye increases heterogonically relative to the number
in the dorsal lobe.
The curve on the right denotes heterozygotes between bar and wild-type (full eye). In the curve
on the left, x denotes homozygous ultra-bar and heterozygotes between ultra-bar and bar ; o,
heterozygotes between ultra-bar and wild-type ;• , homozygous bar.
k for all is close to 1-5 ; 6 is lowered by admixture of the wild-type gene, and is at its maximum
in pure ultra-bar (i.e. rises with decreasing absolute size of the eye). It is clear that facet-formation
must begin ontogenetically in the dorsal lobe.
22
CONSTANT DIFFERENTIAL GROWTH-RATIOS 23
Dimensions (in inches) of Parts of the Bill and Casque of the
Hornbill Antheracoceros malaganus (from E. Banks, unpublished)
$$
<?
Length of gape along
curve ....
3-5
3-8
4-25
4'25
5-o
5*3
5-6
5-8
Length of casque
(straight) .
*
*
2-6
3-2
3-55
4.6
5-2
6*7
Height of casque
above bill .
o-5
O-Q
I-O
0-9
1-4
i-6
i-5
1-85
Height of bill proper
1-2
1*2
i-5
1-25
i-5
i-75
1-6
2-05
* Indistinguishable from bill.
Both in length and height the casque is highly heterogonic
relative to the length and /or the height of the bill proper.
■ 50
/
100
- 20
20
1
50
100
200 mm,
1 ' '
Fig. 13. — Increase of interocular distance with carapace width in the crabs
Cancer (i) and Eriphia (3), and of carapace length with carapace width in
Cancer (2) ; logarithmic plotting.
k for interocular distance, in Cancer 0-70, in Eriphia 0-76.
3 mm.
10mm.
10
20mm
100
e
Fig. 14. — Change of proportions in the crab, Carcinus maenas.
A— D, outline of carapace and of thoracic ganglion in four specimens of different absolute size,
to show negative heterogony of interocular breadth {k = 0-85), and of thoracic ganglion (k= 06).
The carapace lengths have been made the same for all: actually they were 3-1, 109, 30-0 and
71-0 mm. respectively.
E — H, ommatidia of the same four specimens, all drawn to the same absolute scale.
k for ommatidial diameter = 0-32.
24
CONSTANT DIFFERENTIAL GROWTH-RATIOS 25
mm
5
.0
I'
I'
10 20
carapace breadth
50 mm.
Fig. 15. — Diameter of the thoracic ganglion against carapace breadth in crabs :
(1) Pachygrapsus ; (2) Carcinus ; logarithmic plotting.
k in both cases about o-6. (After Teissier, 193 1 : in the original paper, the Carcinus curve is
erroneously ascribed to Pachygrapsus and vice versa.)
Since the above was written, the important paper of Teissier
(1931) has provided numerous fresh examples. These we may
summarize in tabular form (and see Figs. 13-20).
Animal
y = organ
x = standard to which
organ is compared
Value of k
Mealworm, Tenebrio mo-
weight of moult-
total fresh weight
o-8
litor larva
ed skin or cara-
pace
Water-beetle, Dytiscus
, , , ,
>) > t
o-8
marginalis, larva
Water-boatman, Noto-
11 11
)> it
10
necta glauca
Shore-crab, Carcinus
11 11
j » ,,
10
maenas
Crab, Pachygrapsus raar-
11 11
11 > 1
10
moratus
Stagbeetle, Lucanus cer-
weight of desic-
weight of desic-
2-0
vus $
cated mandibles
cated elytra
>i 11 11
weight of desic-
» > M
i-5
cated head
11 11 11
weight of desic-
cated legs
,.
10
1.
mean diameter
length of elytron
10
eye
11 11 11
maximum breadth
head
,.
20
mgr.
100
50
•w 20
I
i
C. 10
•§
* 5
A
?
12 5 10 20 50 100 200 mgr.
body - weight
Fig. i6. — Water-content against body-weight in the larval mealworm, Tenebrio.
Solid line, fresh weight ; dotted line, dry weight ; logarithmic plotting.
k (fresh weight), 0-975 ; (dry weight), 0-92.
mar
Tig,
Ho
0,5
§
0,2
,6
12 5 10 20 50 100 mgr.
body - wetight
Fig. 17. — Total nitrogen against body-weight (solid line, fresh weight ; dotted
line, dry weight) in the larval mealworm, Tenebrio ; logarithmic plotting.
k for fresh weight, 0965 ; for dry weight, o-9r.
26
CONSTANT DIFFERENTIAL GROWTH-RATIOS 27
Animal
y = organ
x = standard to which
organ is compared
Value of k
Stick insect, Dixippus
length prothorax
total length
I-O
morosus
it it tt
length head
it ti
0-71
tt 11 11
breadth head
> » 1 1
0-71
it 11 11
diameter eye
breadth head
11 11
0-48
Mealworm, Tenebrio
ty total weight
o-95
molitor, larva
Crab, Carcinus maenas
interocular
breadth
carapace breath
0-85
}* tt a
carapace length
11 11
about i-o
Pachygrapsus mar-
carapace length
It 11
about i-o
moratus
ts II II
interocular
breadth
11 11
0-85
Eriphia spinifrons
> 1 it
11
0-76
Cancer pagurus .
11 11
11
0-70
Water-boatman, Noto-
diameter of a
total length ^
0-46
necta glauca
single ommati-
dium
Cockroach, Blatta ori-
11 11
11 a
0-36
entalis
Stick insect, Dixippus
11 > >
11 a
0-37
morosus
Crab, Carcinus maenas
11 11
carapace breadth
0-32
,, Pachygrapsus mar-
11 11
a a
0-47
moratus
Crayfish, Potamobius as-
11 it
length
0-40
tacus
Crab, Carcinus maenas
diameter thoracic
ganglion
carapace breadth
about o-6
Pachygrapsus mar-
11 ti
tt n
o-6
moratus
Insect larva, Chaoborus
diameter cerebral
body length
o-6
crystallensis
ganglion
a a a
mean diameter
abdominal
ganglia
tt a
o-6
Stick insect, Dixippus
diameter cerebral
total length
o-6
morosus
ganglion
>» a a
mean diameter
thoracic ganglia
a a
„ o-6
a a a
mean diameter
abdominal
ganglia
a a
o-6
Mealworm, Tenebrio mo-
mean diameter
total length
o-6
litor larva
abdominal
ganglia
Water-boatman, Noto-
diameter 2nd
11 11
„ o-6
necta glauca
thor. ganglion
Insect larva, Chaoborus
mean diameter
body length
0-4
crystallensis
nucleus, nerve-
cells
mgr.
0,5
0,2
0,1
-tr
^
o
<:
^
0,05
J?
-to
3
0,02
0,01
12 5 10 20 50 100 200 mgr.
body - weight
Fig. i 8. — Total phosphorus against body- weight in the larval mealworm,
Tenebrio ; logarithmic plotting.
k for fresh weight, 1-03, later 1-08 ; for dry weight, 0-975, later 1-02.
cod.
500
HO
^ 200
I
o 100
1
50
5 10 20 50 100 mgr.
body - weight
FlG. 19. — Heat of combustion against body-weight in the larval mealworm,
Tenebrio ; logarithmic plotting.
k for fresh weight, 1-07 ; for dry weight, 1-02.
28
CONSTANT DIFFERENTIAL GROWTH-RATIOS 29
Animal
Mealworm, Tenebrio
molitor larva
Wax-moth, Galleria mell
onella, larva
y = organ
x = standard to which
organ is compared
total fat
fat
phos-
phos-
growth
1-07
i-o6
1-04
i-oi
1 00
098
103 0-975
(i-o8 1-02)
I 03
o-975
I-OO
o-945
o-975
0-92
0-965
o-gi
0-87
0-82
0-83
o-8o
0-82
078
o-8i
0-76
1 07
1 02
about o-8
about 0-9
total carbon
dry weight
dry weight,
removed
total phosphorus,
until late in lar-
val life
(total phosphorus,
end of larval
life)
protein nitrogen
fresh weight
total water
total nitrogen
extractives
lipidic
phorus
nucleic
phorus
ash
total heat of com-
bustion
Oxygen consump-
tion at rest and
fasting
Oxygen consump-
tion under nor-
mal conditions,
but on a ration
permitting only
slow
(flour)
ditto, but fed a
ration permit-
ting normal
growth
total water, early
phase*
total water, late
phase
dry weight, early
phase
dry weight, late
phase
total heat of com-
bustion, early
phase
late phase ' refer to the fact that after a period
of regular differential growth lasting for most of the larval period, there
occurs a short phase of irregularity, denoting rapid change in character of
metabolism, followed by a second regular period resembling the first but with
different quantitative relations.
Value of k
(«) (b)
fresh dry
weight weight
(a)
i-i3
(b)
i-o6
about 0-95
10
i-o
0-96 0-91
i-o
i-oS
i-i3
i-o
i-o
I-OQ
Early phase ' and
30
PROBLEMS OF RELATIVE GROWTH
Animal
y = organ
x = standard to which
organ is compared
Value of k
(a) (b)
(«) (b)
Wax-moth, Galleria mell-
total heat of com-
fresh dry
I-l6 i-oq
onella, larva
bustion, late
phase
total phosphor- \
weight weight
us, early phase 1
total phosphor- j
total water
O-Q
us, late phase /
total fat, early ~\
phase [
total fat, late 1
phase /
dry weight
1-32
dry weight, fat\
removed, early
phase
dry weight, fat ['
»»
0-82
removed, late
phase /
Now it is to be observed that a constant differential growth-
ratio, during some at least of the period of growth, is not
merely an empirical rule found over a large range of organs
and groups, but is what one would expect on a priori grounds.
For it is the biologically simplest method we can conceive of
obtaining the enlargement (or diminution) of an organ.
If an organ is to begin its career small and end it large,
the obvious method is to make it grow at a higher rate than
the rest of the body ; the difference once initiated (by what-
ever physiological means) there is no a priori reason why it
should not be maintained at approximately the same relative
level throughout, since changes which affect the growth of the
body will be expected to have, within narrow limits, a pro-
portionate effect on the growth of the organ (and see p. 6).
We shall later consider certain special cases where growth
of organ and body appear not to be equally affected in cer-
tain circumstances (pp. 200, 259) ; others which show that
the formula for constant differential growth-ratio can only be
an approximation when we are dealing with an organ as a
whole (p. 81) ; and still further facts which rule out some of
the early stages of an organ's development from the operation
of the law (p. 139, seq.). However, both empirical facts and
theoretical considerations warrant us in regarding a constant
differential growth-ratio, operating over a longer or shorter
period of time, as the primary law of the relative growth of
parts, once they have reached the stage of full histological
CONSTANT DIFFERENTIAL GROWTH-RATIOS 31
differentiation. It may (like Boyle's law) prove only to be
an approximation, and to be capable of modification in cer-
tain circumstances ; yet (again like Boyle's law) it may remain
fundamental.
We may now proceed to consider a little more in detail
some of the cases. In the first place, we can utilize our formula
to deduce the moment of onset, in male Uca pugnax, of the
large chela's heterogony. To do this we must first know the
weight of the small claw. This in males is identical in form
mqr.
100
50
I
• 10
k 5
1 2 5 10 20 50 100 200 mgr.
body - weight
Fig. 20. — Increase of water-content with total weight in the larva of the
wax-moth, Galleria ; logarithmic plotting.
k for fresh weight, early phase i-o, late phase i-o ; for dry weight, early phase 0-96, late phase 0-91.
with both claws of females, and does not change its relative
weight with increasing body-size : at all stages it weighs, as
does a single female chela, almost precisely 0-02 of the rest-
of-body weight. If we make the assumption that our formula
for the first phase holds from the first moment of increase of
the large chela, we have simply to extrapolate from our formula
and find the point on the curve at which rest-of-body weight
is fifty times chela weight. This is found to be close to 5 mg.
body- weight, when the chela should weigh o-i mg.
32 PROBLEMS OF RELATIVE GROWTH
We can now proceed to check this deduction. Morgan
(1923A, 1924) has found that the very youngest post-larval
U. pugnax he could obtain have both claws alike, of female
or small type, in both sexes. In this stage, the chelae are
autotomized very readily ; and only when one is thus thrown
off does the other proceed to transform into a large chela.
After this has once happened, the fates of the two chelae are
irreversibly determined, though initially either may become a
large chela through the accident of the other's autotomy.
The moment of determination of the large chela appears
normally to take place very early, during the first or second
instar of post-larval life.
Accordingly, I collected and weighed a number of the
smallest fiddler-crabs to be found on the beach, in which the
sexes could not be determined by casual inspection of either
chela-size or abdomen-shape. Their mean weight was about
6-7 mg. — an excellent approximation to the 5 mg. prophesied
on theoretical considerations.
Then again, we can compare relative growth in different
species of the same genus. Uca minax is much scarcer near
Wood's Hole, and the comparatively few specimens available
were all of a size to be in the second phase of U. pugnax.
However, they yielded one or two interesting results. The
double logarithmic plot of chela against rest-of-body clearly
approximated to a straight line ; but owing to the smaller
number available it was impossible to determine the growth-
coefficient of the chela so accurately. It was, however, cer-
tainly between 1-58 and 1-66 — in other words, almost exactly
the same as that of U. pugnax for the first phase. Either
U. minax has no change in the growth-coefficient of the chela
at or near maturity, or at all periods its chelar growth-coeffi-
cient is higher than in pugnax. U. minax also differs from its
relative in the greater size which it attains ; the biggest
specimens found weighed 17-8 g. as against 3-6 g. for U. pugnax.
Correlated with this, as was to be expected, was the greater rela-
tive weight of the large chela to be found in minax. This in one
specimen amounted to no less than 77 per cent, of rest-of-body
weight, as against a maximum of 65 per cent, in U. pugnax.
It is clear that the large chelae of big specimens of U. minax
must be getting close to their maximum limit of relative size.
A claw as big as the rest of the body would not be very prac-
ticable, and these are already over three-quarters this relative
size. In U. pugnax, where our figures are more accurate,
CONSTANT DIFFERENTIAL GROWTH-RATIOS 33
the large chela has
attained half the
rest-of-body size
when the total weight
is about 1-65 g. If
the animal could grow
to 24 g. (less than
30 per cent, bigger
than the biggest U.
minax) , its large chela
would be the same
weight as all the
rest of it together.
Twenty-four grams
is a very small weight
for many crabs, in-
cluding forms of sim-
ilar semi-terrestrial
and burrowing habits
to Uca, such as Ocy-
poda, yet no species
of fiddler-crab has
grown to a size much
over that of U.
minax. It may thus
be plausibly sug-
gested that the exist-
ence of this continu-
ously high growth-
ratio in the large
claws of the male
fiddler-crabs may
have acted as a
limiting factor in
their size-evolution,
any possible advan-
tage to be obtained
by increase in size
not countervailing to
cause selection to
alter the chela's
growth-mechanism .
Analysis of the
3
3 4 5 6 7 8
carapace length, cm.
A
zzo
2-00
/■BO
1-60
/■40
I 20
100
log carapace length, mm.
120
B
1-60
Fig. 21. — Relative growth of male and female
chelae. (A) in Palaemon carcinus, (B) Palaemon
bengalensis ; logarithmic plotting.
(From data of Kemp, I9i3-X5-)
34 PROBLEMS OF RELATIVE GROWTH
relative growth of homologous organs next shows that
the same organ may behave very differently, as regards
its growth-behaviour, in different forms. For instance, the
chela of male Uca pugnax shows heterogony on one side
of the body only, but shows it throughout all but the first
instar of post-larval life, with a decrease in its growth-coefficient
apparently at the time of sexual maturity. In the spider-
crab Maia squinado (Huxley, 1927 ; and unpublished), both
chelae are heterogonic, but heterogony does not set in until
quite late in life, presumably at sexual maturity, and then
continues till death. The same appears to be the case with the
large prawn Palaemon carcinus (Tazelaar, 1930), though here
the chelipeds are the second and not the first pereiopods ;
and with the spider-crabs of the genus Inachus. But in the
latter the male chelae revert more or less completely to the
female type in the non-breeding season ; this reversion is
much less marked in I. dorsettensis (Shaw, 1928) than in
7. mauritanicus (Smith, 1906A). In various other crabs, and in
lobsters, crayfish and pistol-crabs (Alpheus), both chelae are
heterogonic, but with different growth-coefficients, leading to
the condition of heterochely. Furthermore, the sex-difference
as regards the growth-coefficient of the chela may vary, some
forms having equal positive heterogony in both sexes, others
showing positive heterogony in both sexes, but with a lower
growth-coefficient in the female, and still others showing male
heterogony but female isogony. This variability is particu-
larly well shown in prawns (Palaemonidae) ; in these, further,
the large chela is the second, not the first pereiopod. Finally
in Gammarus chevreuxi, Kunkel and Robertson (1928) have
shown that the marked heterogony of the male gnathopod
begins shortly before sexual maturity and ends shortly after,
its growth being roughly isogonic for the much longer previous
and subsequent periods. (Cf. also birds' wings, p. 263.)
A similar state of affairs is seen in regard to the abdomen
of female Brachyura. This must always be heterogonic for
some part of its development, since it is always broad in the
adult, always narrow and of male type in the young juvenile.
A state of affairs similar to that of the large male chela of Uca is
found in the female abdomen of Carcinus maenas — it is hetero-
gonic (with a late increase in intensity) from the earliest stages
until the end of life. The details for both sexes at all ages are
shown and described in Fig. 7. In female Uca, on the other
hand, while heterogony is initiated at the beginning of post-
B
Fig. 22. — Variation in relative growth in the female abdomen of fiddler-crabs
(Uca pugnax).
(A) Left, medium-sized crab (carapace breadth, 10 mm. ; abdomen, 6 mm., broad and reaching
the bases of the legs). Right, medium-sized crab (carapace breadth, n mm. ; abdomen, 5J mm.,
broad, not reaching the bases of the legs).
(B) Means (A — A) and extremes (B — B and C — C) of relative abdomen breadth in female crabs
between 4 and 14 mm. carapace breadth. There is a clear bimodality of mean abdomen size, with
marked heterogony between 8 and 11 mm. carapace breadth.
35
36
PROBLEMS OF RELATIVE GROWTH
larval life, a state
of equilibrium
(adult female pro-
portions) is event-
ually attained,
when the lateral
margins of the
abdomen have
reached the bases
of the legs (Mor-
gan, 1924 ; Huxley,
1924A) (Fig. 22).
(In passing, it
may be noted that
Morgan was led to
postulate female
intersexuality^ in
this species on
rinding certain ap-
parently mature
females with ab-
domens propor-
tionately narrower
than the full
female type. Hux-
ley, however (I.e.),
was able to show
that these were
merely the ex-
treme minus vari-
ants for the
normal variation
curve of female
abdomen - growth,
and that the sup-
position of inter-
sexuality was un-
called for — an
interesting appli-
cation of the study
of heterogony.)
That the equili-
brium-position is
Carapace
Carapwce
Relative abdomen breadth, %
Fig. 23. — Relative growth in the male and female
abdomen in the spider-crab, Inachus dorsettensis.
(A) Left, mature female abdomen ; right, mature male abdomen.
Abdominal breadth was taken on the 6th segment (n — o ; v — w).
(B) Change of means and extremes of relative abdomen breadth
with increase of carapace length in males and females.
CONSTANT DIFFERENTIAL GROWTH-RATIOS 37
not in any way automatically, and still less mechanically,
brought about when the sides of the abdomen reach the
legs is shown by Pinnotheres, the pea-crab (Atkins, 1926)
8 10 12 14- 16 IS 20 22 ZA-
LtriGTri -Cns,
Fig. 24. — Relative growth in the teleost fish, Orthopristis.
The abscissae represent total length. The ordinates represent (above) the length of the head, trunk
(body), and tail ; the division between head and trunk is taken at the hind end of the opercular bone,
between trunk and tail immediately above, or below, the end of the hindmost median fin ; (below)
maximum depth and width of body.
38 PROBLEMS OF RELATIVE GROWTH
in the adult female of which the abdominal margins far over-
lap the leg-bases.
Then we have the quite different case of Inachus (Fig. 23)
which resembles that of the male gnathopod in Gammarus,
only here all the marked heterogony appears to be achieved
in a single moult-period. And finally we have the fact first
discovered by Geoffrey Smith (1906A), that parasitization of
Inachus with Sacculina, while reducing or abolishing the
heterogony of the male chela, actually increases the growth-
ratio of the female abdomen, to remind us that the differential
growth-ratios are only constant in certain conditions.
Similar relations would doubtless be found for other organs,
but these are the best analysed. They show that the dif-
ferential growth-coefficient of an organ, though it may remain
constant throughout post-larval or post-embryonic life, may
equally well be confined to the beginning, the end, or the
middle (and here sometimes to a very small period) of the
life-history, or may change its value slightly but definitely.
Since, however, it is justifiable to regard isogony as a special
case of heterogony, with growth-coefficiency unity, it remains
true that in all cases the growth-coefficients of parts or organs
remain constant over definite periods, and that these periods
are in the great majority of cases few in number and long in time.
(See Hecht, 1916, on the proportions of fishes for a case of
long-continued isogony in all measured dimensions : Fig. 24.) 1
§ 4. Inconstancy of Form and Constancy of
Form-change
In concluding this chapter, it may be pointed out that the
constancy of growth-ratio over considerable periods of the
life-history in spite of environmental fluctuation, is of very
considerable importance for analytical morphology. Where-
ever it obtains, it implies that the form of an animal, as given
by the proportions of its parts, depends (naturally within the
1 Even here, the isogony is not permanent. Up to a length of 30 cm.
— i.e. about a year old — Kearney (reference in Robbins, Brody et al.,
1928, p. 123) shows that there is heterogony. Hecht further points
out that so far as known, all vertebrates with determinate growth
change their proportions continuously up to the adult phase. It is
only in forms with indeterminate growth like fishes that there exists
a long-continued period with no change in external proportions (though
even here the relative size of the viscera changes). The work of Keys
(p. 259), however, indicates that Hecht's conclusions are riot strictly
accurate. Compare also Olmsted and Baumberger (p. 261).
INCONSTANCY OF FORM 39
limits of normal variation, of which more later) solely upon
its absolute size, not upon the length of time it has taken
to reach that size, nor upon changes in any other external
variable. The modifications of and exceptions to this statement
we shall consider later ; here we can accept it as our first
general rule.
As immediate corollary of this we have the fact that only
animals in which all organs are growing at the same rates
will preserve their form unchanged with increase of size ; and
this is as much as to say that no animal will keep its form
identical while increasing in size, for it appears highly improb-
able that any animal will be found in which some organs do
not grow at a different rate from the body as a whole. And
even if for the moment we stick to external form, and further
if we only consider quite large differences in growth-activity,
we shall find many animals in which, as in the male fiddler-
crab, the only constancy of form is the constancy of its mode
of change. This is less obvious and in some ways less important
in the higher animals (notably almost all mammals and birds
among vertebrates and almost all insects and spiders among
arthropods), in which growth ceases at a definite size, and
there supervenes an adult stage of constant size and often
of long duration. For here we can often afford to consider
only the adult forms, in which the proportions of form have
been fixed by the cessation of growth. It is the limitation
of form at a fixed absolute size which confers this convenience
upon the systematist and the morphologist.
Even here, however, as we shall see in detail later, the rule
has many applications. To take the most obvious case, the
absolute size at which growth ceases may be altered by treat-
ment such as feeding ; in such case, the permanently stunted
individual will approximate in proportions to a normal juvenile
stage, the well-fed, abnormally large specimen will have pro-
portions not met with at all among the normal population of
adults.1 It is, in other words, a mere biological accident that
adult proportions, even in species with limited growth, are
relatively fixed ; and to neglect the fundamental fact of change
1 As a matter of fact, the degree of development (including growth)
of a higher vertebrate appears to be simultaneously dependent upon
at least two variables, size and age. This is well shown by Appleton
(1925, see his chart 3) as regards the degree of ossification in new-
born rabbits ; and by Jackson (1925) for young rodents stunted by
underfeeding. Jackson's results are discussed further in Chapter VI.
See also the work of Adolph (p. 258).
40 PROBLEMS OF RELATIVE GROWTH
of proportions with absolute size, and to proceed as if certain
arithmetic (percentage) proportions were immutable ' charac-
ters ' of the species, may lead to serious error.
But we must remember that the limitation of growth and
the consequent establishment of a small range of stable adult
size is a late and specialized feature in evolution. The majority
of animals show unlimited growth : they continue growing,
though usually at a constantly diminishing rate, until they
die, or in asexually-reproducing forms, until they divide. A
lobster or a plaice may increase its linear dimensions several
fold after the attainment of sexual maturity. In such types,
there is no fixed or adult form ; the change of proportions
continues unabated throughout life, and may be as obvious
during post-maturity as during pre-maturity. An excellent
example of this is provided by the detailed studies of Mrs.
Sexton (1924) on the successive instars of Gammarus chevreuxi,
supplemented by the work of Kunkel and Robertson (1928)
on the same species.1
Even among mammals a change of proportions may con-
tinue throughout life. In the voles (Microtinae) Hinton (1926,
Chap. II, 8-14, Pis. Ill, IV, IX) finds that slow growth occurs
long after the adult state has been arrived at, the epiphyses
of the long bones never uniting. This continuous growth is
accompanied by continuous change of proportions. With
increasing size of the skull, for instance, the rostrum becomes
relatively narrower and slightly longer, the interorbital region
narrower and the molars relatively smaller. Unfortunately
the measurements given do not permit of any accurate state-
ment as to the changes involved, or as to the distribution of
growth-potential in different regions. Here is an interesting
field for the student of relative growth. It would be par-
ticularly interesting to discover whether the relative growth-
rates of tail and parts of skull, limbs, etc., remained the same
after the attainment of sexual maturity as they did before.
It would be easier to investigate this on the limb-segments
than on the skull, which undergoes complex distortions and
curvatures. It would also, of course, be necessary to keep
1 Sexton states that sexual maturity occurs at the seventh instar,
that proportions continue to change for two further instars in the
male, one in the female, but after this no further proportion-changes
occur (though the males at least may increase about 40 per cent, in
length). That this statement is not accurate is shown by Kunkel and
Robertson, whose graphs demonstrate a change in the proportions of
several organs up to the largest sizes found.
INCONSTANCY OF FORM 41
the animals under standard conditions, since the work of
Sumner and of Przibram has shown that increased temperature
causes an increased relative size of appendages in rodents.
It is true the change will not usually be of the same extent
after sexual maturity, for although the changes in absolute
size may be greater between maturity and death than in
the period from the post-embryonic or post-larval phase to
maturity, yet the fraction of total growth which takes place
after maturity is always a good deal less, if measured by the
true criterion, namely the amount of multiplication of initial
size. For the fiddler-crab, for instance, the pre-maturity
multiplicative increase in weight is about 250-fold, the post-
maturity increase about three- to four-fold, though the absolute
(additive) increases are roughly as 1 to 2-5. None the less,
the post-mature alterations may be very considerable. In the
male fiddler-crab, after his attainment of sexual maturity, the
proportion of the weight of the large chela increases from
43 per cent, of rest-of-body weight to nearly 62 per cent. —
an increase of nearly 45 per cent, in relative size.
No two male Uca pugnax have the same proportions unless
they happen to be of the same absolute size : any diagnosis
made on the basis of percentage measurements of chelae (and
also, though much less markedly so, for other organs such
as the pereiopods) would be valueless. But in spite of the
fact that the form of the animal is continually changing, it
does so in an orderly way ; and though percentage values
for the limbs have no diagnostic significance, the constants
in the growth-ratio formula are true specific characters. In
a word, the systematist needs algebra as well as arithmetic
in making any diagnoses based upon the size of parts of the
body.
Note. — S. A. Allen (1894), Amer. Mus. Nat. Hist. Bull., 6, 233, also
finds a progressive change of proportions in a rodent (see p. 40).
Neotoma shows a steady increase of dolichopy and dohchocephaly
with increase of absolute size.
CHAPTER II
THE COEFFICIENT OF CONSTANT GROWTH-
PARTITION ; AND SOME SPECIAL CASES
§ i. The Heterogony of Deer Antlers
THERE are certain special cases so important to a
study of relative growth that they deserve a chapter
to themselves.
The first is that of the antlers of deer. As is well known,
these are shed each year, and replaced the year after by a
totally new growth. Usually, each new growth is larger than
the preceding growths, but in old age, illness, or other especially
unfavourable conditions, the weight (and number of ' points ')
may decrease. An analysis of the normal growth of the
antlers of a number of individual red deer (Cervus elaphus)
and of the factors affecting that growth, is given in Huxley
(1926).
Further, casual inspection is sufficient to indicate that
relative antler-weight increases with absolute body-weight,
as is stressed by Champy (1. c). To obtain quantitative data,
however, was not easy. After much search, I hit on the
papers of Dombrowski (1 889-1 892) published many years ago
in an obscure periodical — the only papers to my knowledge to
contain the body-weights and antler-weights of large numbers
of Red deer and Roe deer. These data, supplemented by
those of Rorig (1901), by scattered cases in the literature, and
by information privately supplied to me by sportsmen, have
now been analysed by me (Huxley, 1927, and 1931). It
appears quite definitely that although there may be much
individual variation even in one locality, and though extraneous
agencies such as the amount of lime in the soil affect relative
antler-weight considerably, yet when the mean of considerable
42
¥-&-
■G
p
f\*l
V
/ Gi
cpw
70 I0O
Kg Body wt
\5Q
20 0 250
Fig. 25. — Antler-weight against body-weight in 527 adult red deer (Cervus
elaphus) ; logarithmic plotting. See Table II.
k, except for the last two points, is close to i-6.
43
44
PROBLEMS OF RELATIVE GROWTH
numbers is taken, the results approximate to the formula for
constant differential growth-ratio.1
This applies to adult ani-
mals. When antler-growth
is taken by age for single
individuals, it will be seen
that the differential growth-
ratio of the antler-weight is
not constant, but declines
steadily with age, being first
about 3-0, and declining to
close to i-o. (When regres-
sion of body-weight occurs,
it appears certain that regres-
sion in antler-weight accom-
panies it, though it cannot
yet be stated whether the
regression is heterogonic.)
It is doubtless affected also
by numerous subsidiary fac-
tors such as abundance of
food and specific dietary
1 A discrepancy occurs as re-
gards the antlers of those beasts
with highest body-weight ; the
weights of these when expressed
as relative (percentage) weights,
fall below those of the body-size
class next below. This appears
to be merely a classificatory
phenomenon. There being con-
siderable individual variation as
to what we may call the par-
tition-coefficient of material be-
tween antlers and body, those
animals with the very largest
body- weights are likely to repre-
sent extreme variants in the
direction of heavy body but
light antlers. Further, and poss-
ibly more important, since body-
weight is extremely variable
owing to fluctuations in amount of fat, most very heavy beasts are
likely to owe their exceptional weight to exceptional nutritive con-
ditions ; and therefore their relative antler-weight will go down rela-
tively to this excess of fat, which is presumably without immediate
significance in determining antler-size.
/
/
/
/
n
/
5
//
tj
/ f
4
/
/ i
t /
/©
3
/
/ J
So'
/
i 2
c
t i
■s
i
So
/
/
1
/
/
t
0-5
r
60
80
ICO 130
Kg. Body wt.
170
Fig. 26. — Red deer, antler-weight
against body-weight at various ages ;
logarithmic plotting.
The antler- and body-weights for the first 7
years of life (in stags from Warnham Park, Sus-
sex) are plotted. The curve (solid line) bends over
and approximates to the straight line curve (dotted
line) for an tier- weight against body- weight in adults
(see Fig. 25).
k begins with a value of 3-0 or over, and declines
to i-6 or under.
HETEROGONY OF DEER ANTLERS
45
TABLE II
Body-weight, Antler-weight, Point-number and Relative Antler-
weight of 527 Red Deer shot in various parts of Europe
(392 collected by Dombrowski ; 10 by Baillie Grohman ;
125 by Huxley. Analysed by Huxley, 1931A)
Arranged by body-weight classes, all of 20-kg. interval (except the
last class, of 40-kg. interval) . Note that for Classes 2 to 8 (comprising
over 90 per cent, of the animals) the relative antler-weight rises steadily
with increasing body-weight, k for antler-weight (except for the
last two classes) is about i-6 ; b = -00162.
kg.
Class body-
weight
No. of
specimens
Mean
body-
weight
kg.
Mean
antler-
weight
kg.
Mean
point-
number
Relative
antler-
weight
per cent.
of body
Relative
pt. no.
• pt. no. n
Vbody-wtJ
I 60- 80
19
74-4
1-64
7-50
2-20
o-ioi
2 80-100
119
93-4
2-03
820
2-17
0-088
3 100-120
106
110-4
3-16
9-81
2-86
0-089
4 120-140
113
130-6
3-96
11-64
303
0-089
5 I40-160
65
1489
478
1 1 74
3-21
0-079
6 160-180
29
1707
6-21
13-10
364
0-077
7 180-200
33
191-1
7-28
14-77
3-8i
0-077
8 200-220
18
2II-8
8-91
I5-4I
4-21
0-073
9 220-240
H
231-7
879
13-62
379
0059
IO 240-280
11
259-1
863
1378
3*33
0-053
TABLE III
Body- weight, Antler-weight, Point-number and Relative Antler-
weight of 405 Roe Deer shot in various parts of Europe
(data from Dombrowski ; analysed by Huxley, 1931A)
k for antler- weight = about 0-57 ; b = -0455.
Class, Body-weight
No. of
specimens
Mean
body-
weight
kg.
Mean
antler-
weight
g-
Mean
point-
number
Relative
antler-
weight
per cent,
of body
1-5 (13-18 kg.)
6-12 (19-25 kg.)
13-15 (26-39 kg.)
127
254
24
16-6
20-9
28-3
225-7
257-0
306-5
5-92
6-05
6-21
1-36
1-23
1-08
46
PROBLEMS OF RELATIVE GROWTH
factors. The reason that the weights for adults fall upon the
line corresponding to a constant growth-ratio with k = about
i-6 appears simply to be that during the decline of the antler's
growth-ratio, a rather narrow range of values for the growth-
coefficient is attained during adult life (see Fig. 26), the great
majority falling between say i-8 and 1-4.
o-4
0-3
SJ
0-2
1 >
/
/
/
f
/ ^
/ S^
js£
s /
_^£L y
^^ /
/
/
/
/
/
/
. L
30
40
Fig. 27.
15 20
Kg Body- wi
-Relative size of antlers in adult Roe-deer (Capreolus caprea).
Solid line, antler-weight against body-weight in 405 Roe-deer ; k = 0-57. Dotted line, prolongation
of corresponding curve for adult Red Deer (see Fig. 25). Logarithmic plotting. See Table III.
Corresponding data for the Roe deer (Capreolus capreolus)
gave what at first sight appeared a quite paradoxical result —
namely a decrease of relative antler-weight with increase of
absolute body- weight among adult males (Fig. 27). There is
thus negative heterogony of the antlers, with a growth-co-
efficient of about 0-57. Reflection suggests the probable ex-
planation. There is no reason why the decline in the antler's
growth-coefficient with age should not in another species
proceed much faster than in the Red deer, and reach a stage
HETEROGONY OF DEER ANTLERS
47
where by the attainment of maturity it was normally below
i*o. This purely quantitative difference in the rate of change
with age would suffice to explain the apparently contradictory
results (Fig. 28). As to the biological causes underlying this
Kg. Body wt
o-i
hO-08
0-06
1-0O4
8 10 10 30 40 60 80 100 150 200
Fig. 28. — Diagram to compare probable method of antler-growth in Red and
Roe-deer ; logarithmic plotting.
X — X, the curve for adult red deer (see Fig. 25). A — A and B — B, probable curves for individual
antler-growth with age in a small and a large specimen respectively. Y — Y, the curve for adult
roe-door (see Fig. 27). C — C, probable curve for individual antler-growth in a typical roe-deer
specimen, rising at first more rapidly, but then sinking much lower than the corresponding curve for
red deer.
48
PROBLEMS OF RELATIVE GROWTH
quantitative difference we can only speculate : it would seem
probable that the Red deer type of slow decrease, with positive
heterogony throughout, is the normal course of events in
Cervidae, but that it was for some reason biologically desirable
for the Roe deer to have small antlers.
12 13 14
Fig. 29.
Solid line, body-weight against age in 212 male red-deer from Wamham Park. Dotted line, antler-
weight against age in a smaller and selected group of stags from the same locality. The thin continuous
lines below the curve for antler-weight represent diagrammatically the actual growth and shedding
of the antlers year by year. The fact that the antler curve inflects later than that for body-weight
is probably due to the antlers being from a selected group of beasts, of size above the average.
After this digression, we will return to the general problem
involved in the growth-ratio of the antlers. We have seen
that the apparent constancy of their growth-ratio, obtained
by plotting antler-weight against body-weight in adults, is
shown to be a particular consequence of the steady decline of
THE COEFFICIENT OF GROWTH-PARTITION 49
individual growth-ratio with age. But even this does not
exhaust the complexity of the phenomenon. The curve for
age-change of growth-ratio is obtained by plotting the weights
of fully-formed antlers of known age against body-weight for
the same age. It will be at once clear that the actual growth-
ratio of the antlers can never be the same as that thus obtained,
but must always be higher (Fig. 29). For the points on the
curve are those which would be obtained if the antler grew
with a constant differential growth-coefficient from its incep-
tion ; whereas actually, it has to begin its growth anew each
year from zero.
Now this is of considerable importance, since it indicates
that it is not necessarily the actual rate of growth which is
regulated in accordance with our formula, but the limitation
of the total amount of growth achieved. What our results tell
us is that at any given body-size the total amount of material
which can be incorporated in the organ is proportional to the
body-size raised to a power (the exact value of the power also
varying with age). The mechanism of this relation is at
present obscure. We do not know whether the total bulk
of material in the body imposes the relation directly, which
is unlikely ; whether some substance is formed in the body
in this particular quantitative relation, as an exponential
function of body-weight, and the final size of the organ is then
directly proportional to the amount of this substance ; or
whether there be after all a true constant differential growth-
ratio between organ and body, determined by some peculiarity
of the organ, but that this growth-ratio represents a limiting
value, higher values being possible and indeed necessary
whenever the relative size of the organ is below its limiting
amount. The last supposition is perhaps the most probable,
on the close analogy with regeneration (see below) , but experi-
ment alone can decide the point.
§ 2. The Coefficient of Constant Growth-partition
In any case, to speak simply of growth-coefficients in such a
case is misleading ; yet we require a term for the exponent
of body-size according to which relative organ-size is limited.
Two terms are possible — either coefficient of growth-limitation,
or else growth-partition coefficient ; I shall adopt the latter.1
1 Since writing this passage, I find that Robb (1929) had previously
suggested the same idea of growth-partition, which has later been
adopted by Twitty and Schwind (193 1).
4
50
PROBLEMS OF RELATIVE GROWTH
In general, it would appear that the existence of a growth-
partition coefficient is the most fundamental fact in consider-
ing relative growth of parts, and that when true constant
growth-coefficients or constant differential growth-ratios are
found, they represent special limiting cases of this more general
conception.
Let us now consider three further examples which support
this conclusion. I have mentioned regeneration. I shall deal
with this more fully in a later chapter. Here it suffices to
recall the fact that in an animal capable of full regeneration,
any organ, heterogonic or not, will, after amputation, be
restored in favourable conditions to its normal proportionate
II III IV V VI VII VIII IX x XI XII
50
40
3 0 -
20-
10
126
Fig. 30. — Decrease of growth-coefficient during regeneration in the legs of
Sphodromantis bioculata.
The abscissae represent moult-stages. The ordinates are growth-quotients : i.e. the ratio of the
length of the leg at a given moult to its length at the preceding moult. The dotted line represents
the mean growth-quotient (1-26 = V 2) for normally-growing limbs. The solid line is the curve for
a middle-leg amputated before the Illrd moult (mean of 3 specimens).
size. In other words, during the process of regeneration its
growth-ratio will be much higher than normal, and will gradu-
ally sink until it reaches the normal value, at which it will
then continue. This emphasizes the generally accepted idea
that regeneration is simply a special case of growth, and
furthermore makes it clear that here at least the normal growth-
ratio of an organ merely represents a limiting value. Thus,
as was suggested above with regard to deer-antlers, relative
size of organs appears to be determined in the first instance
as an equilibrium between amount of material in the organ
and amount of material in the body, the equilibrium being
determined according to our general formula y = bxk. If the
THE COEFFICIENT OF GROWTH-PARTITION 51
equilibrium be upset, regulation towards the equilibrium
position will occur during later growth. The particular
mechanism by which the equilibrium is attained does concern
growth-ratio ; the more the organ is below equilibrium-size,
the higher will be its growth-ratio.1 (See Fig. 30.)
These conclusions are supported by various lines of evidence.
In the first place, in cases of grafting of organs we should
expect the organ of a young animal grafted on to an older
and larger animal to be accelerated in its growth until it
reached a size prescribed by its growth-partition coefficient,
and the organ of an older animal grafted on to a younger
and smaller animal to be correspondingly retarded in its
growth. For the first, we may turn to the results of Wachs
(1914). When he inserted the lens of a young Urodele larva
into the eye of an older larva from which the lens had been
previously removed, the small lens was accelerated in its
growth. For the second, as well as the first, we have an
example in the work of Twitty (1930). Here cross-trans-
plantation was made between larvae of Ambly stoma tigrinum
and A. punctatum. The latter species grows much more slowly
than the former. Twitty removed the eye of a punctatum
larva and replaced it by one of the same size from a tigrinum
larva ; owing to the higher growth-rate of tigrinum, the age
of the donor was much less than that of the host. Even
when the host was fed only minimally, the grafted eye now
increased in size much more rapidly than that of the host :
in one case it increased 50 per cent, in diameter while the
host remained stationary in length. (See p. 197.)
The converse experiment consisted in removing the eye
from a tigrinum larva and engrafting in its place an eye of the
same size from a considerably older punctatum larva. In this
case, the grafted eye made very slow growth. Here the rate
of growth could be compared with that made by punctatum
eyes grafted into A. tigrinum during the embryonic period.
1 In some cases at least the change in growth-ratio will occur accord-
ing to the law enunciated for Sphodromantis by Przibram, 1917 :
When Z is the normal final length of the regenerating limb, n the
length after amputation, r its length at the beginning of a given period
of time t, R the length at the end of the time t, Va the normal
coefficient of increase of the limb between one moult and the next,
then Z — n — y = — — ; and the growth-partition coefficient
represents the limiting value of the growth-ratio when equilibrium is
established.
52
PROBLEMS OF RELATIVE GROWTH
It can at once be seen (Fig. 31) that the growth of the older
eyes slowly approaches the normal growth-curve (see also
Figs. 85 to 87, and especially 88).
2.0
mm.
i.TS
ISO
UJ
>
UJ
IL
O
arm
UJ
i-
uJ
Z
d
1.0
•is
20 30 HO SO 60
BODY LENGTH
70
£0
90 mm
Fig. 31. — Regulation of eye-size in eyes of one species of Amblystoma grafted
on to another.
Dotted lines, curves for two cases when embryonic eyes were grafted to a host of the same stage
of development as the donor. Solid lines, curves for two cases when eyes were taken from an older
larva and grafted on to a younger host larva ; in this case the eyes hardly grow at all until they
reach the correct relative size.
We thus are driven to the conclusion that though the eyes
of the two species have different specific growth-intensities,
and therefore different coefficients of growth-partition when
both are present in the same body, there is for each body-size
THE COEFFICIENT OF GROWTH-PARTITION 53
carapace length, mm
13 16 20
a characteristic eye-size towards the attainment of which the
rate of eye-growth is regulated.1
In general, as is well known,
the rate of regeneration is higher
the more material is removed.
This fits in with the ideas here
presented and with the concep-
tions of Przibram (1. c), but it
throws into relief the very real
difference between the idea of
a constant differential growth-
ratio and a constant coefficient
of growth-limitation. Let us
consider a Planarian worm, in
which according to Abeloos (I.e.)
the trunk grows heterogonically
with reference to the head.
During normal growth the rela-
tion is one of a constant differ-
ential growth-ratio. But dur-
ing regeneration, the more is
cut off, the more rapidly regen-
eration takes place : i.e. the
smaller the fraction of the body
left, the more rapid is the
growth-ratio of the regenerate.
What is constant is the final
partition-coefficient between
head-material and trunk-
material ; the normal constant
differential growth-ratio is the
special case of growth during
which the partition-coefficient
is always of this limiting value.
Finally, the case of Inachus,
investigated by G. Smith (1906A)
whose data have been further
analysed by me (unpublished)
also supports this view-point. At Naples, /. mauritanicus , the
species of Inachus studied by Smith, shows three forms — ' low '
1 Further details as to the specific growth-intensities of eyes and
other organs when heteroplastically transplanted are recorded in
Chapter VI.
Fig. 32.- — Chela breadth against
carapace length in the male of the
spider-crab, Inachus mauritanicus ;
logarithmic plotting. (See Table
IIIa.)
The ' low ' males (below about 14 mm. cara-
pace length) and the ' high ' males (above about
20 mm.) fall on a single curve with diminishing
growth-coefficient (mean value of k over 2-3 ;
for ' low ' males, about 2-6; for 'high' males,
about 1-4). Between these sizes the chela
regresses to a narrow female type, then enlarg-
ing again with a very high growth-coefficient.
(Constructed from the data of G. W. Smith, 1906A.)
54
PROBLEMS OF RELATIVE GROWTH
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HOLOMETABOLOUS INSECTS 55
males, of small size with relatively small but male-type chelae,
' high ' males of large size, with relatively large male-type
chelae, and those of intermediate size, which have extremely
small, female-type chelae.
When chela-size is plotted double-logarithmically against
body-size, it is found that the means for the ' low ' and ' high '
males fall on two segments of a single simple curve, thus con-
firming Smith's view that these two types are merely breeding
males in their first and second seasons respectively, and that
those with female-type chelae are males in the non-breeding
phase, during which the secondary sexual characters of their
chelae have regressed to the female or neuter type.
Further, on the double logarithmic plot, the curve for these
intermediate males first actually declines, and then mounts
very steeply until it meets the prolongation of the straight-line
curve for the low males, upon which it bends over and con-
tinues as the line for the ' high ' males. In other words, after
the regression period, the growth-ratio of the claw is much
higher than normal, but becomes normal as soon as the theor-
etical equilibrium-size is reached.1 (Table IIIa and Fig. 32).
It is also seen that the frequency for chela-breadth is bimodal :
this will be discussed in § 5.
§ 3. HETEROGONY IN HOLOMETABOLOUS INSECTS
A somewhat different set of special cases is that provided
by holometabolous insects. Many of these possess organs
(usually of secondary sexual character, and these usually
in the male sex), which increase in relative size with increase
of absolute size of body. The most familiar of these are the
mandibles of the stag-beetles (Lucanidae) and the ' horns ',
cephalic or thoracic or both, of various other beetles such as
the Dynastidae ; but Champy (1. c.) has collected numerous
other examples, ranging from antennae (e.g. Acanthocinus :
Champy, 1924, p. 167) and forelegs, to the ' tail' on the hind
wing of Papilios and the swollen segments of the hindlegs in
certain Hemiptera, such as Anoplocnemis (Champy, 1924,
p. 173). See Figs. 33, 34, 91.
Analysis shows (Huxley, 1927 and 1931) that the relation
between the dimensions of the organ and the body here too
1 It is interesting to find that in I. dorsettensis, studied by Shaw
(1928), the regression towards female type in the non-breeding season,
though present, appears to be much less marked.
56
HOLOMETABOLOUS INSECTS
57
approximates closely to the formula for a constant differential
growth-ratio (Table IV, Fig. 35). Again, however, there can be
no growth-ratio in the literal sense in which we have found
Fig. 34. — Heterogony of the ' tail ' in the male of the swallow-tail butterfly,
Papilio dardanus (the heterogony is stated by Champy not to occur in species
in which the ' tail ' occurs in both sexes).
it apply, e.g., to the partition of growth-potential between
the large chela of Uca and the rest of the body. There cannot
be, for the simple reason that in holometabolous insects the
organ, as regards its imaginal characters, is not formed until
58
PROBLEMS OF RELATIVE GROWTH
the pupal instar, to emerge at the final moult in its definite
shape and size. And as there are no further moults, it is
incapable of further growth or form-change.
We are thus driven to suppose either that all the processes
connected with the organ's heterogonic growth are confined
to a very brief period, presumably just before and just after
the moult from last
larval in star to pupa ;
or else that, although
the visible growth of
the organ is confined
to this short period, it
depends for its amount
on some substance
whose chemical ac-
cumulation during the
larval phase has had
a constant differential
growth -coefficient
relative to body-
weight (see e.g. Teiss-
ier, 1 93 1, for a confir-
mation of this latter
possibility).
As we shall see later
in considering dimor-
phism (p. 68) the for-
mer hypothesis is the
more probable ; but in
any case we have, as
in deer's antlers, the
fact that the amount
of growth attained is
proportional to body-
size raised to a power,
the value of the power
30 40 50607080
Total length, mm.
Fig. 35. — Relative growth of male mandibles
in three species of stag-beetles (Lucanidae).
-f , Lucanus lunifer ; X , L. cervus ; 0, Cyclommatus laran-
dus. ' Total length ' is true total length for Cyclommatus ; for
the others it is represented by (elytron length + mandible
length). All the curves inflect at large absolute sizes (see
text) ; for the remainder of the curves k is about 1-6 for
L. lunifer, 2-3 for L. cervus, and nearly 2-0 for C. tarandus.
being equivalent to
that of the constant differential growth-ratio in cases where
visible growth is continuous over long periods. Thus the true
growth-ratio of the organ during its short growth-period is far
more rapid than indicated by the value found for the ' growth-
coefficient ' by the method of comparing organ and body
at different absolute sizes. We are, in fact, again in the same
HOLOMETABOLOUS INSECTS
59
predicament as with the deer's antlers, the chelae of male
Inachus, or any heterogonic organ which is regenerating, and
are driven to think of a limiting factor to growth, which we
have defined as the growth-partition coefficient.
TABLE IV
Mandibles in Lucanidae
(a) Cyclommatus tarandus
(data from Dudich, 1923 : ana-
lysed Huxley, 1927)
2
k
x
20-38
24-01
26-38
27-76
29-65
32-20
33-n
35-oi
36-13
37-32
38-44
39-26
41*34
43-22
45-51
46-32
47-28
48-40
50-04
5i-5o
52-50
54-23
56-01
62-06
66-06
69-00
74-00
178
1-97, b
y
3-88
5'3i
6-33
7-32
8-17
9-73
10-71
11-49
12-08
12-73
14-11
14-70
15-84
17-39
18-83
19-19
19-92
20-79
21-53
22-54
23-25
23-96
25-38
28-49
30-69
32-00
34-50
= just over o-oi
(b) Lticanus cervus
(data from Bateson and Brindley,
1892 : analysed Huxley, 1927)
x
(3i-o)
38-65
40-50
42-55
45 -oo
46-93
49-18
53-6o
2 = 48
y
(6-o)
7-75
9-00
io-oo
II-20
n-86
12-82
14-40
k = about 2-3
Lucanus lunifer
(data and analysis, Huxley, 1927 )
X
y
(38-6)
(16-8)
42-4
19-9
45-6
22-4
49-3
24-5
5i-9
26-0
57-3
28-8
2 =
18
k =
aboul
: i-55
Figures in brackets ( ) indicate insufficient number of individuals
in class.
In L. cervus and L. lunifer, x = 'total' length = (elytron length +
mandible length), mm.
In C. tarandus, x = true total length = (body length + mandible
length), mm.
y — mandible length, mm.
k = growth-partition coefficient : the values given do not hold at
high values of x (see text).
60 PROBLEMS OF RELATIVE GROWTH
There is a further point to consider in regard to heterogony
in holometabolous forms. In other organisms — a fiddler-crab,
for example — the growing system is an open one, in that it
is continuously taking in food as it grows. The beetle or
other holometabolous insect, however, during most of the
period when the form of its adult organs is being laid down,
is a closed system, taking in no further food, but depending
on accumulated reserves and on the material derived from
the breaking down of larval organs. This has two conse-
quences for our problem. In the first place, in other forms
the fairest comparison of relative size would seem to be between
heterogonic organ and rest-of-body, since the size of the
ingestive and digestive systems are functions of the size of
the rest of the body, not of total size, as may easily be realized
by reference to the fiddler-crab. Here two large specimens,
male and female respectively, of equal rest-of-body weight
and therefore presumably equal-sized alimentary systems, will
be of very different total weight, since in the female either
chela will weigh only 2 per cent, of the rest-of-body, while in
the male the large chela may weigh up to 70 per cent, or more.
But in a stag-beetle, for example, the conditions are quite
different. Larvae of both sexes have jaws and guts of the
same relative size. A male and a female larva of the same
total size will have the same amount of reserve material, but
during the pre-pupal and pupal period, say 1 per cent, of
this is converted into imaginal female mandible, while perhaps
10 or 15 per cent, has to be converted into imaginal male
mandible. Since the amount of reserve material is here the
important factor for growth, it is total bulk, and not rest-of-
body bulk, which should be here used as the standard against
which to plot the bulk of the heterogonic organ (see Huxley,
1931c).
This is a minor methodological point ; but it has further
consequences. To continue the example of stag-beetles, we
should accordingly expect, in an exceptionally large male
specimen where the theoretical relative size of the mandibles
would be huge (as long as the rest of the body in some speci-
mens of Cyclommatus tarandus : Dudich, 1923 ; see Fig. 91), that
during the longer time necessary to lay down this large organ,
the limited reserve-supply would come to an end, used up by
other competing organs, and therefore that the organ would
fall below the theoretical size expected on the formula for a
constant coefficient of growth-partition. Thus, owing to the
POLYMORPHISM IN NEUTER INSECTS 61
limitation of raw material due to the system being a closed
one, we should expect, from a certain absolute size upwards,
the actual values for the size of the heterogonic organ to fall
progressively more and more below the theoretically expected
value. And this is what we actually find. When organ-size
is plotted against total size on a double logarithmic grid, the
first part of the curve is a good approximation to a straight
line, but the end portion curves over so as to be concave to
the x-axis. This is so for all cases so far investigated, includ-
ing the mandibles of three species of two genera of stag-beetles,
the horn of Xylotrupes, the heads of polymorphic neuter ants,
etc.1; Figs. 35, 37, 92.
§ 4. Heterogony and Polymorphism in Neuter Social
Insects
We shall later note some further complications introduced
into the situation by the fact of moulting. Here we may refer
to the particularly interesting case, just mentioned, of poly-
morphic neuter ants. It is well known that in what are
apparently the more primitive examples of such polymorphism,
there is an unbroken array from smallest to largest neuters,
the continuous series being quite arbitrarily divided up into
' worker minimae ', ' worker mediae ', and so on up to ' soldier
mediae ' and ' soldier maximae ' ; and some myrmecologists
have introduced even more elaborate terms (see Wheeler,
1920). Now these series are invariably characterized by a
relative increase of head- and especially mandible-size with
an absolute increase of total size. Measurements of the
weights of head and rest-of-body in species of two genera
(the huge Camponotus gigas, from Borneo ; and a driver ant
of the genus Anomma from Africa) show that, over the major
portion of the size-range, the formula for constant growth-
partition coefficient is nicely adhered to 2 (Huxley, 1927 ;
Huxley and Bush unpublished) (Table IVa and Fig. 37).
1 Teissier, 193 1 (p. 97), using weight and not linear measure, shows
in his Fig. 20 no curving over of this type for the mandibles of Lucanus
cervus. This might mean that my interpretation is wrong, and that
mechanical reasons are interfering with great growth in length rather
than nutritive reasons with growth in mass. On the other hand, the
curvature in my material does not begin until elytron-length 33 mm.,
and Teissier has hardly any specimens as large as this.
2 The Anomma curve bends over at high sizes, as described for stag-
beetle mandibles, etc. This may presumably be accounted for as
suggested earlier in this chapter. But the formula is also not obeyed
62 PROBLEMS OF RELATIVE GROWTH
TABLE IVa
Anomma nigricans
Data from Huxley and Bush (unpublished)
Analysed in Huxley (1927)
(1383) (165-3)
176-0 223-7
220-3 300-2
272-7 427-6
324-2 567-3
367-5 661-9
2 = 267
k = about 1-55 (after first 2 points)
x = abdomen-length
y = head-breadth
(in arbitrary units)
Figures in brackets ( ) indicate insufficient numbers in class.
Camponotus gigas
Data from E. Banks (unpublished)
Analysed in Huxley (1927)
x y
75-0 9-6
in-5 24-3
241-0 82-0
346-8 142-0
S =357
ft — about i-6 (after first point)
x = total weight, mg.
y = head-weight, mg.
The obvious suggestion is that these series of workers and
soldiers do actually represent nothing more than a series of
size-forms of a single genetic type, possessing a mechanism
for heterogony of the mandible and head. The difference
from the other holometabolous cases hitherto considered is that
the absolute size-range is much greater, and that the differ-
ences in size can only be supposed to be brought about by
definite treatment of the larvae by their nurses, the largest
types being fed to the limit, the smallest types being deprived
of food, and so forced to pupate, while still quite small larvae.
That enormous differences in size may be produced by cutting
down or cutting off the food supply of insect larvae is estab-
lished through experimental work on blowflies (Smirnov and
at very small absolute sizes, where the double-logarithmic curve is
distorted in the opposite sense, with concavity upwards : the meaning
of this, if not merely statistical, is unknown.
POLYMORPHISM IN NEUTER INSECTS
63
Zhelochovstev, 1926), housefiies (Herms, 1928), Drosophila
(Gause, 1931), etc. And the postulated differential treatment
of different worker larvae by their nurses is well within the
known range of complexity of ant behaviour (see also Emery,
1921).
Fig. 36.-
-Increase of relative size of head with absolute size of body in the
neuters of the ant, Pheidole instabilis.
That the effect of size-changes may be differential is also
known ; e.g. Eigenbrodt (1930) finds that the increase of total
size in Drosophila caused by low temperature is accompanied by
a decrease in wing-size, which is somewhat greater for breadth
-zoo
(A) Heterogony of the head in neuter
ants; logarithmic plotting. © — ©,
Anomma nigricans, head-breadth against
abdomen-breadth (in arbitrary units : scale
below and to the right). H h Cam-
ponotus gigas, head-weight against total
weight, in mg. (scale above and to the
left).
(B) Relative head-size, Anomma nigri-
cans. Increase in percentage head-breadth
with increase in absolute abdomen-breadth.
(In this figure the classes into which the
specimens were classified were delimited
on the basis of [abdomen-breadth + head-
breadth].)
140 ZOO
%
180
170
160
140
130
Anomma
nigricans
B
200 250 300
cubdomen breadth
350
Fig. 37,
-Graph showing the increase of size of head with absolute size of
body in neuter ants.
64
POLYMORPHISM IN NEUTER INSECTS 65
than for length ; Alpatov (1930) rinds the same phenomenon.
Smirnov and Zhelochovstev (I.e.) for the blowfly Calliphora find
a differential effect in different regions of the wing when total size
is reduced by cutting off larval food-supplies after a given time.
On the other hand, Alpatov (1. c.) in flies of the same species
reduced in size by depriving larvae of food before they reached
full growth, also finds a decrease in relative breadth of the
wings, showing that absolute size is not the only factor deter-
mining proportions.
When, as in some more specialized forms, the neuter series
is not continuous, but there exist only two or a few fairly
sharply defined types — e.g. only large and relatively large-
headed soldier and small and relatively small-headed worker —
we need only suppose a slight specialization of the nurses'
behaviour. An adumbration of this is seen in the fact, elicited
by unpublished work of Miss Edmonds, of Sydney, that even
in forms with a continuous series of neuters, the frequency
curve for body-size is definitely multimodal (see also Palen-
itschenko, 1927).
It is to be hoped that those familiar with the technique
of rearing ants in captivity will attack this problem experi-
mentally. If my suggestion be verified, it will materially
simplify the evolutionary problems connected with the poly-
morphism of ant neuters, for instead of having to postulate a
large number of genetically distinct types, we need only assume
one single genetic type of neuter, but provided with a heter-
ogenic mechanism for head-growth, which will produce all
the different head-types as secondary by-products of the
animals' absolute size.
In termites, the case is somewhat different.1 Within the
soldier caste of primitive termites, we do sometimes meet
with phenomena which appear to be quite parallel with what
I have discussed for ants — considerable variation in absolute
body-size accompanied by a progressive alteration in relative
head- and jaw-size. This is well shown in the figures given by
Heath (1927) ; see Fig. 38. Heath also (see his p. 402) demon-
strates fairly conclusively the interesting fact that the poly-
morphism is due to the presence of forms which have gone
through different numbers of moults ; a fact which strengthens
our hypothesis that there is but one genetic type with a heter-
1 For much of the information concerning termites I am indebted
to Professor A. E. Emerson, of Chicago University, to whom I should
here like to express my thanks.
5
66
PROBLEMS OF RELATIVE GROWTH
ogonic mechanism as regards head-size. The different soldier
forms are discontinuous as regards size and head-proportions,
which provides us with a further case of di- or poly-morphism
due to a combination of heterogony with moulting (§ 5).
Reference may also be made to some of the results given in
the important paper by Kalshoven (1930), and in that by
Hare (1931). Such a method of arriving at polymorphism of
neuters would, of course, be impossible in the holometabolous
ants. (See also the work of Light, p. 258.)
There are however other termite cases in which my hypo-
thesis would not seem to apply. For in-
stance, A canthotermes acanthothorax (Sjostedt,
1925 : his Fig. 21, p. 61) has certainly two
qualitatively different types of soldiers, which
it would appear necessary to regard as differ-
ent genetic types, or at least as produced by
qualitative differences in feeding. Sjostedt
himself figures no less than five kinds of
soldiers, the largest number of types decribed
for any species of Termite. Two of these,
his Nos. 4 and 5, would appear to be growth-
forms of one main type. To inspection, his
Nos. 1-3 look as if they were growth-forms
of another qualitatively different type ; but
Dr. Emerson informs me that he has found
that the smaller forms (Sjostedt's Nos. 2 and
3) are both infested with an insect larva which
inhabits the head, and disproportionately
reduces its size ; these forms are not found
in nests from which the parasites are absent.
The differential retardation of the head might
be due to the existence of a true heterogony-
mechanism for the head, which is not nor-
mally manifested, but is revealed when the parasite produces
general size-reduction of the imago ; or it might be due merely
to the fact that the head is the seat of the infestation.1
The difference between worker and soldier would appear
1 In passing, we may note that the effects of such parasites may
be very striking ; e.g. in Termes gilvus, Silvester (1926) figures extra-
ordinary qualitative changes in shape of head and jaws, as well as a
general size-reduction and a highly disproportionate reduction in jaw-
size, as result of the presence of a similar parasite in the head. And
see, for a discussion of the corresponding problem in ants, Wheeler
(1928) and Vandel (1930).
Fig. 38. — Heads of
largest and smallest
workers in a colony
of the termite Ter-
mopsis angusticollis,
showing heterogony
and change of pro-
portions with in-
crease of absolute
size.
POLYMORPHISM IN NEUTER INSECTS 67
to be of another nature. Recent work such as that of Emerson
(1926), John (1925), Heath (1927, 1928), etc., makes it highly-
probable that in all primitive termites, the forms which do the
work of the colony are not a distinct caste, but are the juvenile
forms of the soldiers. There is thus a marked heterogony
of the head and jaws between the last worker instar and the
first soldier instar (we have already seen that there may be
more than one soldier instar : Heath, 1927 : see also Hare,
I93I-)
In more specialized forms, however, while some of the
workers are juvenile soldiers, a true worker caste, distinguish-
able by large size and different proportions, also exists
(Emerson, 1926, etc.) It would be extremely interesting to
measure the head and body of these two types throughout
their growth.
These results, combined with the fact that in some primitive
forms, fertile soldiers are met with (Heath, 1928 ; Imms, 1920),
and that soldiers with wing-buds are not unknown, whereas
no true workers are known ever to be fertile or to possess
wing-buds, indicates that workers have been derived from
soldiers by a suppression of their final development into the
normal big-jawed type — that, in fact, they are neotenic. This
neoteny, however, possibly at first facultative, must at least
in higher forms have been fixed as a constant caste-character-
istic, either by differential feeding or perhaps more probably
by some genetic mechanism (see Thompson, 1917).
This has a bearing on our problem, since the main feature
in the evolution of the worker from the soldier would simply
be the delay in the onset of the head's heterogony, a delay
which, when it exceeded the time to the final instar, would
eventually lead to the total absence of soldier characteristics.
A less degree of delay would give rise to forms of intermediate
soldier-worker type ; these are known to exist in various
genera, e.g. Armitermes and related genera.
Thus, although heterogony appears to play its part in the
origin of polymorphism both in ants and termites, its role is
different in the two groups. In ants, the control of absolute
size through the feeding of the larvae appears to be the main
method. In termites true heterogonic polymorphism is rare,
and when present seems due to irregularity of moult-number ;
but neoteny due to postponement of the onset of heterogony
also plays a role in the differentiation of castes.
68 PROBLEMS OF RELATIVE GROWTH
§ 5. Heterogony, Moulting, and Dimorphism
Finally, certain quite different special problems arise from
the interaction of constant differential growth-ratios with the
characteristic Arthropod process of moulting. It will be clear,
since growth in a typical Arthropod only occurs between the
shedding of one exoskeleton and the hardening of the next,
that in the life-history of any single specimen the theoretical
growth-curve relating organ-size with body-size will never be
realized. Instead, growth of both organ and rest-of-body will
take place in a series of jumps, but the points thus arrived at
will all lie on the theoretical curve.
Although moulting in Crustacea and some insects tends to
take place at each doubling of weight (Przibram, 1930), the
large variations encountered, together with the variation in
the initial post-larval weight, are often sufficient to obscure
any recurrent modality in the sizes at which moulting occurs.
In a large population of such species, moulting thus occurs
at random at any size, and measurements of a heterogonic
organ whose growth-ratio is constant over long periods accord-
ingly fall on to a continuous curve when made on such a
population.
The state of affairs, however, is entirely different if (a) the
organ's high growth-coefficient is confined to one or a few
instars and (b) is initiated in a more or less constant phase
of an instar — e.g. always immediately after moulting. As
the simplest case, let us take one in which the high growth-
ratio lasts but one instar, as occurs with the female abdomen
of Inachus (Shaw, 1928). When the data are grouped into
classes by body-size, and then simply the means of the various
size-classes taken, a curve is obtained which merely indicates
two periods of approximate isogony, separated by a short phase
of intensive heterogony (Figs. 22, 23). But when frequency-
curves for relative abdomen-size are prepared for each body-
size class, the true state of affairs is revealed (Fig. 39). All
female Inachus are then seen to fall into one or other of two
sharply non-overlapping groups as regards relative abdomen-
size. Those below a certain absolute body-size all have narrow
abdomens, those above another higher body-size all have
broad abdomens. The curves for the short intervening range
of body-size, however, are all bimodal, some individuals having
narrow, others broad abdomens, but none being of intermediate
type. When the curves for all the classes are summed, we
Inachus ?
22
21
20
19
18
17
16
15
14
13
12
II
10
- T T
T T
T T .
T T
35 40 45 50 55 60 65 70 75 80°/
-40-45-50-55 -60-65-70-75-80-85 /t>
30
- r T T
29
28
27
26
- T T
2 b
—
24
-
23
-
22
T T T T
21
20
19
-
T , .
18
-
17
_
T
T _
16
-
11
T
T
14
-
13
-
T
12
-
11
-
10
-
9
8
3^-
T
1 1 1 — — 1 —
— I 1 1 1 1 1 1 1 1 '
0 1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16
Fig. 39. — Dimor-
phism due to moult-
ing in (A) female
abdomen (breadth
of 6th segment)
and (B) male chela
(breadth of propus)
in the spider-crab,
Inachus dorsetten-
sis.
The values are relative,
expressed as percentages
of carapace length. The
frequencies for each value
of abdomen and chela
are given for various
values of carapace length,
and also for all specimens
taken_together.
70 PROBLEMS OF RELATIVE GROWTH
again of course obtain a bimodal curve for the whole population
(Fig- 39)-
This can only mean that the change from narrow to broad
type is effected during a single instar; and further that it
must be initiated at a relatively constant phase of the instar,
for otherwise there would be greater variability in the broad-
type abdomen than is actually found. Finally, the fact that
the bimodal curves are found over a considerable range of
body-size indicates that there is not one particular serial
number of moult, or one limited body-size, at which the
transformation (which presumably is associated with sexual
maturity) is initiated. If the high growth-ratio of the abdomen
had extended over two instars, we should have had a trimodal
instead of a bimodal curve ; but the more instars over which
it extended and the greater the variation in the body-size at
the initiation of heterogony, the more obscured would the
curve's multimodality become, owing to the variation in the
amount of growth at each moult, which becomes cumulative
with the increase in the number of moults concerned.
A similar but less-marked bimodality occurs in this species
as regards chela-size in males, and is doubtless to be explained
in the same way ; see also Table IIIa, p. 54, for a similar
phenomenon in /. mauritanicus .
The important point to notice is that the unusual and
apparently abnormal fact of dimorphism in one sex has here
been brought about by a combination of the two normal and
common processes of moulting and heterogony.
It will at once be seen that this type of dimorphism is
quite different from that noted by G. Smith (1. c.) which we
have just considered in regard to the chelae of males of the
same genus. The ' low ' and ' high ' males in this case were
simply the groups of small and large body-size in what would
have been a case of continuous heterogony resembling that
of the chela of male Uca, if it had not been for the intercalation
of a non-breeding period in which the claw reverted to female
type. This dimorphism has nothing to do with moulting,
but is due to an interruption of a long-continued heterogony
by a non-breeding phase ; while that of the female abdomen
of the same genus is due to the restriction of heterogony to
a single moult-period. Both, however, are dimorphisms of
developmental origin, and have nothing in common with
genetic dimorphisms like those of ' diphasic ' mammals or
birds such as arctic fox, certain squirrels, herons, owls, etc.,
butterflies. They also differ
HETEROGONY, MOULTING, AND DIMORPHISM 71
or the genetic polymorphism of the females of various mimetic
from the environmental di-
morphisms, the best-known
case of which is that of the
wet-season and dry-season
forms of certain tropical
butterflies, where the differ-
ence between the two types
is elicited by different envir-
onmental conditions bring-
ing out different expres-
sions of the same gene-
complex.
O
I I
J_Jj.
■ ■ -- 1 I I
J-..
All
L
a
I I- 1 --T .
,_L
17
16
15
14
m m
8
13
,+
4
12
h
+JL
3-5 4-5
—r-
5-5
6-5 7-5 8
forceps-length, mm
■5 9-5 10-0
10
10 11 12131415l61718Mm
Fig. 40. — Bimodality and hetero-
gony of the male forceps in the
earwig, Fovficula.
Left, absolute frequency of different forceps-
length at various body-lengths in a random
sample of 445 specimens. Right, logarithmic
plot of means of ' high ' forceps {h — h), ' low '
forceps (I — I), and both together (dotted),
against body-length in 1,519 specimens, k
for all specimens, about i-6.
Can we apply the results found in the abdomen of female
spider-crabs to the classical cases of dimorphism in holometa-
bolous insects — those of the earwig Forficula and the beetle
72
PROBLEMS OF RELATIVE GROWTH
Xylotrupes (Bateson and Brindley, 1892) ? I think that we
can. Let us begin with the clearer-cut case of Forficula, in
which there is a definite dimorphism of the male forceps, but
no dimorphism of female forceps, or of the body-size of either
sex.1
TABLE V
Measurements of 1,519 Earwigs collected by Djakonov : ana-
lysed in Huxley, 1927B, Table I (see Fig. 40)
fcuD
,d"
-t->
Low type
High type
d
d
,
in +-»
</i ^j
.— 1
cue
a c
>>
in
en
O
0 3
.0 d
d
6
d
"rt
cxd
££
0 .
O !-c
4-1 4J
> -
6
0 -
0 >-i
0
d.5
£>
d
rt
O
d ko
■43-d
0
S 5P
■BtS
0
<r.
(A
0
6
H
rt d
5 M
05 .s
H
-2 b£
0) d
055
In
0)
10
5
_
370
5
370
37'°
.
. .
o-o
II
40
4-15
38
404
367
2
6-25
56-8
5-o
12
160
4-4i
139
4'°5
33-8
21
6-8i
56-8
i3-i
13
363
5-24
219
4-18
32-2
I44
6-86
52-8
397
14
481
5-80
216
4-26
3o-4
265
7-06
50-4
55-i
15
326
6-6i
69
4-32
28-8
257
7-23
48-2
78-8
16
129
7-28
15
477
29-9
II4
7-61
47-6
88-4
17
15
8-17
■ — ■
■ ■
15
8-17
48-1
ioo-o
Total
1,519
I3-87
5-80
701
4-19
—
818
7-17
—
53-9
Analysis of Djakonov's results (Huxley, 1927B) has shown
that when the data are tabulated by body-size, the mean
1 Recently Kuhl (1928) has attempted to show that the dimorphism
of the forceps of male earwigs is apparent only, due to unconscious
selection of largest and smallest specimens in collecting. This, how-
ever, does not account for the monomorphism of male body-size or
of female forceps ; and in any case is quite unable to account for the
degree of dimorphism found by Bateson and Brindley, Djakonov (1925),
etc. If apparent dimorphism is so easily produced by such means,
existing collections should demonstrate it for large numbers of species ;
instead it is quite exceptional.
It would appear that the absence of, or slight tendency to, bimodality
shown in Kuhl's material (his pp. 362-3) is to be accounted for by the
almost total absence of ' high ' forms in his material, which again is
to be correlated with low body-size. The maximum forceps-lengths
in his four samples are 6-5, 6-5, 7-5 and 8-o mm. respectively. As
my figures show, the ' high ' mode only occurs at 7-5 to 8-o mm. In
Djakonov's material the maximum forceps-length is 10 mm., and
Brindley (referred to on my p. 313) records a maximum of 12-25 mm. !
HETEROGONY, MOULTING, AND DIMORPHISM 73
values for forceps-length against body-length give a good
approximation to the formula for constant differential growth-
ratio between forceps-length and body-length. When, how-
ever, frequency curves for forceps-length are plotted for each
class separately, a situation is revealed analogous to that for
Inachus female abdomen. The smallest specimens show uni-
modal curves, their forceps being all of ' low ' type ; the
largest, also uni-
modal curves,
with forceps all of
' high ' type ; and
the medium-sized
show b i m o d a 1
curves, the num-
bers of low forceps
diminishing and
high forceps in-
creasing with in-
crease of body-
size. It should be
noted that low and
high types, as with
Inachus chela and
abdomen, d i ff e r
only in size ; fur-
ther, that in this
case there is slight
overlapping of the
two modal types
(Figs. 40, 42.)
Przibram (1927)
has suggested that
the bimodality is
due to some male
earwigs having
one or more extra moults ; but his contentions, as they stand,
will not explain the facts, for they should give bimodality of
body-length as well as of forceps, and they do not take account
of the overlap of ' low ' and ' high ' types, between which there
is neither qualitative nor quantitative difference, but merely a
modal distinction depending on their frequency distribution.
Furthermore, it is difficult to see how more than one extra
moult can ever occur without leading to tri- or multi-modality.
Y
'
/
,C
c
; L
5?
/
">
/ /
O
J,
A /
e
'b
A
yx
L
M
L
M
L
/
M
2 n — 1 n n -j- 1
Moult-stages {and Body-size)
Fig.
— Diagram to show the possible origin of
dimorphism of the male forceps in earwigs, Forjicula.
LM, LM . . . sizes of larval forceps at successive moults ;
growth isogonic. X, point at which heterogonic growth of forceps
is supposed to be initiated. If the change to the imago occurs at
the nth moult, the modal level of adult forceps-size is at A — A
('low' males) ; if at moult (n + i), the forceps, after continuing
of larval type for one more instar (B — B), would reach the modal
size-level, C — C (' high ' males).
74 PROBLEMS OF RELATIVE GROWTH
The hypothesis of an extra moult may however be used to
account for the facts, along the following lines. The processes
responsible for the heterogony of the male forceps begin
operating, we will suppose, late in larval life, and usually at
a fairly definite phase of an instar. They cannot, however,
be expressed in the form and size of male-type forceps until
the imaginal instar is reached. This may be reached either
at the next moult after the initiation of the heterogonic process,
or only at the second moult. The result is that the heterogonic
processes have either been operating for less than one instar,
or for a period more than double as long, with resultant
bimodality. Variation in the intensity of the heterogony,
and in the time after moulting at which it is initiated, will
bring about the overlap of high and low ; there is, however,
no qualitative difference to be expected between the two
forceps-forms (Fig. 41).
The interesting fact discovered by Djakonov (1. c.) should
be noted, namely that unfavourable nutritive conditions
cause a decrease in mean and modal body-size, both for the
population as a whole, and for the ' low ' and ' high ' groups
considered separately. As regards forceps-length, however,
the main effect is to cause a shift in the distribution of the
forceps, there being fewer high-type and more low-type, but
without alteration of either modal value (Fig. 42). Furthermore,
though in unfavourable conditions there are fewer large indi-
viduals in both high and low series, yet for classes of the same
mean body-size, whether we consider the population as a
whole or the high and low groups separately, the mean forceps-
size is actually greater for the animals which have grown up
in the unfavourable conditions.
This apparently paradoxical fact may be explained in terms
of our previous discussion. The unfavourable conditions
reduce the total body-size. But the initial growth-ratio of
the heterogonic forceps, starting at a late moult, will always
be the same ; it is only its final value which will tend to a
limit imposed by the growth-partition coefficient. For most
body-sizes therefore we should expect that the absolute size
of the forceps would depend only on the time for which it had
grown, not on the body-size at which it began to grow, and
therefore with stunting of total bulk, a given body-size will
show an absolutely as well as a relatively greater forceps-
size. It is, further, quite possible that the equilibrium-size
and final theoretical growth-ratio is in practice never attained,
HETEROGONY, MOULTING AND DIMORPHISM 75
the imaginal stage with its cessation of growth supervening
before this point is reached. (This is perhaps supported by
the fact that the double-logarithmic plot for all forceps taken
7
to
20
15
10
& 5
K
I 020
15
10
5
X
I s /
D,
\
\
,*
/c
—
/ /
/ /
D\
\c
-
D,
/--% \
/
/c
\
\c
1 -<
•** s
1 1
D
1
1
"-J^
-j 1
10 11 12 13 14 15 16 17 18
body -length, mm.
•St
5
35 4-0 5-0 60 7-0
forceps-length, mm.
B
80
90 9-5
Fig. 42.— Different types of changes produced by unfavourable conditions
(D — D) as against favourable conditions (C — C) on (A) body-length and
(B) forceps-length in male earwigs. The modal body-length is reduced ; the
frequency of forceps of low or high modal lengths is altered, but the modal
length remain the same.
76 PROBLEMS OF RELATIVE GROWTH
together does not curve over downwards at high body-sizes,
as with various other holometabolous forms.) If so, then
with earwigs within the known range of body-size, those with
stunted body will always have larger forceps, though this
should not hold if we could produce individuals of a much
larger body-size.
A further peculiarity of the curves is that those for high
and low forms taken separately (Fig. 40) both show a pheno-
menon the reverse of that found in most holometabolous
insects — namely that they are concave upwards at their upper
ends ; in addition, they are concave downwards at their
lower ends, indicating a rapid growth-coefficient at either
end. This may, perhaps, be accounted for if we suppose that
growth-ratio is accelerated during the period just before and
just after a moult. The high-sloping early portion of the ' low '
curve would be due to the normal initial high growth-ratio
of the forceps-rudiment at the first onset of heterogony, and
would comprise those individuals which started their forceps-
heterogony late in the instar prior to the imago. The high-
sloping end portion of the same curve would include those
which started their heterogony relatively early in the same
instar. But those in which heterogony was initiated towards
the middle of the instar would only have one period of accelera-
tion at the end of a period of slow growth-ratio, and therefore
would show less change of forceps-size for a given increase of
body-size. If an extra instar is added, the rapid growth-
periods are repeated, with corresponding results on the curve.
This is, of course, purely speculative, but may serve as the
basis for further work.1
The case of Xylotrupes (Huxley, 1927c) has been less
thoroughly analysed. It presents various complications, most
notable being a tendency to trimodality over a certain range
of body-size.
In Lucanidae (Huxley, 1931c) analysis of Dudich's paper
(1923) and other data show several interesting facts. First,
that in Cyclommatus the range of male body-length (without
mandibles) is much greater than in other stag-beetles, being
about as much as the mean body-size, whereas in Lucanus
cervus and L. lunifer it is less than half the mean. (The ratio
of largest to smallest body-length in Dudich's Cyclommatus
specimens is 2-47 ; in Bateson and Brindley's Lucanus cervus,
1 Most of the suggestions here advanced concerning Forficula
modify or extend those put forward in my paper on the subject (1. a).
HETEROGONY, MOULTING, AND DIMORPHISM 77
the corresponding ratio for elytron-length is i-6.) Secondly,
the range of the male's body-size is much greater than that
Fig. 43. — Heterogony and moulting of the 3rd pair of legs of the mite, Analges
accentor inns.
Above : (1) large male with relatively huge 3rd legs ; (2) small male with 3rd legs of the same
relative size as in females ; (3) small male in which the onset of heterogony has apparently taken
place earlier, giving slightly enlarged 3rd legs. Below : the 3rd legs of the 3 forms, showing details.
of the female's, though both start at the same minimum size
(16 mm.) : the largest males have bodies 40 mm. long, the
78 PROBLEMS OF RELATIVE GROWTH
largest females only 25 mm. As the mandibles of the male
are highly heterogonic (ranging up to 88 per cent, of body-
length), the disparity in total size is even greater. This differ-
ence would appear only to be obtained on the basis of possible
extra moults in the male. In this form the frequency curve
for all mandibles is again multimodal, with two main and one
or more subsidiary modes. But the multimodality is not so
well defined when curves for separate body-size classes are
plotted. There is also a slight tendency to multimodality or
at least irregularity in the male body-length frequency curve.
No multimodality is apparent in the Lucanus curves. Thus
here again the insect with the presumption of a facultative
extra moult is characterized by multimodality of heterogonic
organ.
A case of male dimorphism in the Acarine mite Analges
accentorinus which would appear to be due to similar causes
has been described by Jucci (1924). Here the large adult
males are characterized by a great hypertrophy of the third
pair of limbs. There further exist small but also adult males
whose third limbs are almost identical with those of the female.
From the scale drawings given by Jucci, we can say that the
size-difference between the two forms is very close to what
we would expect (on the supposition that bulk is approxim-
ately doubled at each instar) if the larger had had one more
instar in its development than the smaller (Fig. 43).
In addition, occasional small males are found with slightly
enlarged third limbs : these would be specimens in which the
onset of heterogony had taken place slightly earlier than
usual. The case is thus very similar to the earwig, save that
the normal ' low ' type is more like the female than in Forficula.
Having concluded this survey of special cases, I shall in the
next chapter pass to a more detailed analysis of the empirical
laws of relative growth in heterogonic organs or regions.
CHAPTER III
GROWTH-CENTRES AND GROWTH-
GRADIENTS
§ i. Growth-gradients within Single Organs
SO far we have only dealt with the question of relative
growth in whole organs or regions of the body ; and in
our first chapter we have found an approximation to a
simple mathematical formulation of relative growth, which
we have called the law of constant differential growth-ratio.
This, as we have further seen, is what we should have expected
if we had worked on the problem a priori. It teaches us the
striking fact that relative growth-rates of different parts of
the body may stay constant over long periods of growth,
which clearly is important as a contribution to the problem
of form co-ordination and the orderliness of form-change, but
it sheds little light upon any aspect of the growth-process
itself.
In this chapter, however, we shall deal with certain further
empirical laws or rules which will, I think, have to be taken
into careful consideration in any future investigation of the
biology and physiology of growth. It is of some interest that
these rules, which to my mind constitute the most important
part of any contribution made by me to the study of relative
growth, emerged quite incidentally out of the investigations
on differential growth-rates. In studying these latter, I had
a perfectly clear-cut aim — to see whether change of propor-
tions could be envisaged as the result of any simple laws of
relative growth. But of the growth-gradients to be discussed
in this chapter, I had no suspicion : their existence thrust
itself upon me as a new empirical fact, any explanation of
which is for the moment entirely problematical. I say a new
empirical fact, for although D'Arcy Thompson (1. c.) had
already adumbrated a similar view, for one thing he had not
fully generalized it or pursued its consequences to their limit,
and for another, it was new to me, as I had not at first grasped
79
8o
PROBLEMS OF RELATIVE GROWTH
ilea pugnax
6 large claw
i>
the full implications of his ideas, which only became clear on
re-reading his book after obtaining certain empirical results
for myself.
The starting-point of these investigations was afforded by
the fact, obvious to simple inspection, that whereas the pro-
portions of the separate joints of the female-type chelae in
the sexes of Uca do not change appreciably during growth,
those of the joints of the large or male-type chela do change,
and very markedly.
The most obvious alter-
ation is a relative in-
crease of the size of the
propus with absolute
increase of the size of
the whole chela. How-
ever, when the weights
o f different chela-
regions were accurately
determined, it was
found that there existed
within the limb what
we may call a growth-
gradient, the distal re-
gion (chela + propus)
having the highest rela-
tive growth-rate, the
central region (carpus)
the next highest , and the
basal region, nearest the
breaking-joint, (merus
+ part of ischium) the
lowest, although its
relative growth-rate
was still above that of
the body.
If we put this crudely into graphic form, using growth-
coefficients as ordinates and spacing the different regions
arbitrarily along the *-axis, we obtain a curve representing
the distribution, along the main axis of the limb, of what we
may for brevity's sake, without introducing any theoretical
ideas, speak of as growth-potential. This curve is inclined to
the horizontal, and is therefore the graphic representation of
a growth-gradient within the appendage, the inclination of the
carpus
Fig. 44. — Graph to show different relative
growth-rates of different parts of the large
claw of the fiddler-crab, Uca pugnax.
Weights in mg. of dactylus + propus (o, scale on right)
and merus + ischius (+, scale on left) against weights of
carpus ; logarithmic plotting. The distal region shows
positive heterogony (k about 1-05) relative to the inter-
mediate region (carpus) ; the proximal region shows nega-
tive heterogony {k about 0-9). (See Table VI.)
GROWTH-GRADIENTS 81
curve representing the steepness of the gradient, or in other
words the difference in absolute growth-potential between the
two ends of the gradient (as measured by growth-coefficients,
which for our present purpose afford the only comparable
standard for measuring intensity of growth-potential in a
number of different regions or forms) in relation to length of
the gradient — i.e. the relative length of the appendage. Such
a graph, however, is as I say only a crude representation of
the true growth-gradient. The most obvious reason for this
is the impossibility of assigning fixed points along the abscissa-
axis to the several joints, since the very fact of their differential
growth is causing their centres (or ends) to shift differentially
with increase in absolute size.
And secondly, we have the difficulty that the values we
have obtained for the growth-coefficients are merely mean
values for large regions of the organ, whereas if the idea of
a growth-gradient be really justified, we should expect a pro-
gressive change of the growth-coefficient from point to point
along the axis, even within the limits of a single joint — a
theoretical consideration supported by certain actual evidence
in other forms (pp. 98, 261-2).
The full solution of the problem, so as to obtain a quantita-
tively accurate picture of the graded change in growth-potential
along an organ, will be a matter of considerable difficulty,
partly owing to the formal difficulties arising from the con-
stant change of the relative size of the parts measured, partly
owing to the practical difficulty of finding sufficient distinctive
points within the limits of a region such as the segment of a
limb, on which to take measurements to determine the detailed
form of the gradient empirically and not by mere extrapolation
from a few mean values.
Finally, there is still another difficulty. As pointed out
to me by Mr. J. B. S. Haldane, if y be the value of the
weight (or linear measurement) of the organ as a whole, and
if y 1}y2, . . . yn be the corresponding values of its constituent
joints or segments, then if y = bxk be a correct expression
for the limb as a whole, then yx = b^1, y2 = b2xkt, and so
forth cannot be accurate expressions for the separate parts
(or vice versa), since the sum of the several expressions for
the parts will not exactly fit the expression for the whole.
On the other hand, within certain limits of the value of k,
the discrepancy will only be slight, and we are justified, from
the actual figures obtained, in taking an expression of the
6
82
PROBLEMS OF RELATIVE GROWTH
above general form as giving a close approximation to the truth,
and therefore in using the values of the growth-coefficients
(k) as obtained from this type of expression as standards of
growth-intensity.
UCA PUGNAX c? .
Region
TABLE VI
Mean Weights (mg.) of Three Regions of Large
Chela (57 Specimens)
Distal (dactylus + propus)
Intermediate (carpus)
Proximal (merus + ischium
to breaking- joint)
Class 1
2
3
4
5
114
179
222
280
344
17-5
257
33-9
40-6
507
28-8
42-2
557
63-1
76-9
515
70-8
103
TABLE VII
Growth-coefficients of Different Regions of the Large Male
Chelae in Different Crustacea (based on Huxley, 1927 and
unpublished, and dean, unpublished, analysis of kemp)
Units of Measurement.
Species
Weight, relative to total : Uca pug-
chela weight (distal ! nax
to breaking-joint)
Maia
squinado
Weight, relative to
body-weight
Length, relative to total Palaemon
cheliped length (dis
tal to breaking-joint)
rudis
Merus +
ischium
Carpus
1
0-89
0-97
o-8i
o-93
I-5I
172
ischium
1-04
0-83
merus
1-02
1-19
2-22 2*IO
Investigations are now in progress which have for their aim
the clearing up of some of the practical and theoretical diffi-
culties in the way of obtaining a quantitatively accurate
picture of the growth-gradient within an organ. Until these
have been completed, I shall here content myself with estab-
lishing the fact that a gradient of some sort exists ; and
in the graphic representation of growth-gradients I shall
GROWTH-GRADIENTS & GROWTH-INTENSITY 83
arbitrarily represent the centres of homologous regions as
equidistant along the abscissa axis, and shall use the ^-values,
as obtained from the formula for constant differential growth-
ratios, as reasonable approximations for the values of growth-
intensity.
§ 2. Steepness of Growth-gradient within an Organ
and Growth-intensity of the Organ as a whole
The fact of a growth-gradient once established for the
heterogonic chela of Uca, the next step was to see if similar
growth-gradients occurred in other heterogonic organs. This
proved to be the case. We will first take other examples
from Crustacean appendages. The weights of the separate
joints of the chela (five of them distal to the breaking- joint)
were taken for both male and female Maia squinado (Huxley,
-10
c //
£ 10
OS
0
§>*«
•a-"" ^
merits +
ischium
carpus
axis of chela. —
dactylus
+ propus
Fig. 45. — Growth- gradient in the large chela (x) of the fiddler-crab, Uca,
(©) of the spider-crab, Maia.
The growth-coefficients of the different regions are here taken relative to the total weight of the
chela distal to the breaking-point, not, as in Fig. 44, to the carpus-weight.
1927, and unpublished). It was found that while the pro-
portionate weight of the joints of the female chela remained
approximately constant within the limits of variation at all
absolute sizes, — i.e. their growth-coefficients relative to the
chela as a whole, like that of the chela relative to rest-of-
body, were all = i-o, — those of the male chela during its period
of heterogony were all greater than unity and were arranged
in a regular growth-gradient. This was double in form, with
high point in the propus, a slight fall towards the tip (dactylus),
and a more rapid and more prolonged fall towards the body.
The high point of the gradient we will call the growth-centre.
84
PROBLEMS OF RELATIVE GROWTH
Presumably the growth-gradient in the large chela of Uca was
of the same form, but this was not apparent, owing to the
propus and dactylus having been lumped together for purposes
of measurement. (See Table VII ; Fig. 45.)
The gradient is steeper in the chela of Maia than in that
of Uca. This appears to be due to the fact that though the
c? chela of Maia begins its period of heterogony much later
10 20 30 40 50 60 70 BO 90 100
120
140 160 160 200
I
<*.
, 30
■u
g 28
I
<0 26
° 24
.-&■
O
•&■
10 20 30 40 50 60
80 100 120 140
cheliped length, mm.
160
180
200
Fig. 46. — Changes in relative length of different segments of the cheliped of
the prawn, Palaemon ritdis, with increase of cheliped length.
Solid lines, males ; dotted lines, females. From above downwards : merus; dactylus; ischium (0),
and propus (v->). The ischium decreases markedly, the merus and the carpus are almost constant;
the propus increases markedly (growth-centre), the dactylus slightly.
in life than that of Uca, and therefore never attains the same
enormous relative size, yet during this period, its growth-
coefficient is higher than that of Uca (about 1-85 as against
i-6 in Uca's first phase and 1-3 in its second). It would appear
natural that the greater is the growth-coefficient of an organ
as a whole, the steeper will be the growth-gradient of its parts ;
this is confirmed by all the evidence so far collected.
GROWTH-GRADIENTS & GROWTH-INTENSITY 85
The only references I can find to these striking changes in
the proportions of heterogonic Crustacean limbs are those of
Kemp and his fellow-workers (Kemp, 1913, 1914, 1915 ; Hen-
derson and Mathai, 1910). But they only draw attention to
the change in percentage length of the joints, and have not
proved the existence of constant differential growth-ratios, or
propounded the idea of a growth-gradient. The figures for
Palaemon rudis have, however, been analysed by Miss I. Dean
(unpublished) and in both series show a definite growth-
gradient with growth-centre in the propus (Table VII ; Fig. 46).
In general, measurements of crustacean limbs show that
wherever there is marked heterogony, there is a comparatively
steep growth-gradient within the limb, with well-marked
growth-centre near the tip (apparently always in the propus) ;
when the heterogony is only slight, the growth-gradient is far
less steep, and its centre usually near the middle of the limb :
in most cases examined either in the merus or carpus. Thus,
although the joints of the small (female-type) chela of male
Uca do not, to simple inspection, appear to alter in propor-
tionate size, measurement shows that in respect at least of
linear dimension, they do so, albeit slightly ; the growth-
centre here is in the carpus (Huxley and Callow). (It will
later be shown that precisely similar relations hold for those
brachyuran abdomens which have been measured.)
Benazzi (1929) has given results on the regeneration of the
limbs of the larva of the dragonfly Aesckna grandis from which
it can be calculated that for regeneration during two instars,
the growth-coefficients (k) of femur, tibia and tarsus, relative
to the sum of the three parts, are as follows : femur, 1-12 ;
tibia, 1-02 ; tarsus, 072. There is thus a regeneration-gradient
in the limb with high point proximally.
Whereas the gradients of most organs appear to be of
the simple form above described, there are some of unusual
type in which part of the organ has growth-coefficients
above unity, the rest below unity. This is the case with
the first antennae of certain copepods (Seymour Sewell,
1929) (see Fig. 47). There is a positive growth-centre at
the eighth or ninth segment, and a negative growth-centre
at the extreme tip. The change from positive to negative
heterogony of the segments, relative to total antenna-length,
occurs close to the joint between the eighteenth and nine-
teenth segments. It is worth noting that when a hinge is
developed in the male's grasping antenna, it is formed at this
86
PROBLEMS OF RELATIVE GROWTH
joint. In almost all cases relative growth again falls off
steadily on the proximal side of the positive growth-centre.1
The first and sometimes a few more of the basal segments
usually exhibit negative heterogony, but occasionally show a
very low positive heterogony. The first antenna as a whole
shows a slight negative heterogony relative to total length
(I.e., p. g).2
We may suggest that it is biologically desirable for the
terminal portion of the antenna to decrease, the proximal
10 II 12 13 /■» IS 16 17 IB 19 20 21 22 23 24 2i
antennal segments
Fig. 47. — Change in proportions of segments of first antenna of Copepods
during growth. Constructed from the data of Seymour Sewell, 1929. He
gives the proportionate sizes of the antenna segments relative to total antenna
length at various sizes.
The graph gives the percentage change in proportionate size between the smallest and largest
stages measured (the segments have been grouped as indicated, and the means taken for the groups).
In every case the growth-rate of the basal region is low, usually negatively heterogonic ; there is a
centre of maximum growth at the 8th or 9th segment, and a centre of minimum growth at the distal
end. The transition from positive to negative heterogony occurs at about the 18th or 19th segment.
X Nannocalanus minor. + Eucalanus subcrassus.
region to increase in relative size. Since changes in relative
size appear to operate by means of growth-gradients, the nega-
tive growth-centre in the tip of the antenna will be connected
1 Seymour Sewell's data for Undulina vulgaris indicate that in this
species, after a low point of no change in proportions in the fourth
segment, the growth-gradient again turns upward as we pass towards
the body, which would give a still more complex growth-gradient.
2 The increase in total body-length, however, is due partly to the
formation of new segments in the growing zone in the sub-terminal
region of the abdomen. As I shall attempt to show in a later chapter,
growth during early stages of the process, during which differentia-
tion from embryonic tissue is actively proceeding, obeys different laws
from those concerned with heterogony of parts which are already
differentiated. It would be better to compare the growth-rate of the
antenna with some definitely-formed part of the body, e.g. cephalo-
thorax-length, in which case it would probably show slight positive
heterogony.
GROWTH-GRADIENTS IN NEGATIVE HETEROGON Y 87
by a continuous growth-gradient with the positive centre in
the region of the ninth segment. Further, we may safely
assume that the copepods, in common with almost all other
animals, show a negative heterogony of the head region :
accordingly this centre of low growth-intensity will again be
connected via a continuous growth-gradient with the positive
growth-centre of the antenna. Since the gradient is of the
same type in both sexes, the functional differentiation of the
terminal region of the male antenna as a clasping organ cannot
have any causal significance in determining the low growth-
rate of this region. On the other hand, the fact that the
heterogony passes from positive to negative at about the
eighteenth or nineteenth segment may have had something to
do with the fixing of the hinge- joint between the clasping
region and the rest of the male antenna at this spot ; such
a suggestion must, however, be regarded for the moment as
purely speculative.
§ 3. Reversal of the Sign of the Growth-gradient in
Negative Heterogony
Those pereiopods which are used as walking legs appear
usually to show slight but distinct positive heterogony, and
to have a definite but slight growth-gradient with centre in
the merus (Bush, 1930). It is of interest that in the actively-
running shore-crab Ocypoda, the young (like the active young
of Ungulates) must be provided from the start with relatively
large legs if their speed is to be sufficient, so that their pereio-
pods show a definite negative heterogony, or decrease in rela-
tive size with increase of absolute size : and that here the
low point of growth, or ' negative growth-centre ', is also in
the merus1 (Cott, I.e.; Huxley, 1931B).
A similar reversal of gradient-sign appears to occur in the
individual development of Ungulates. D'Arcy Thompson (1. c.)
gives a figure (Fig. 48), of the proportions of the foot in ox,
1 This is from length-measurements kindly supplied in answer to a
query of mine by Mr. Cott ; unfortunately, he only had a few speci-
mens available for measurement, and the results, while clearly showing
the merus as the joint of lowest growth-ratio, are not sufficient to
construct a growth-gradient. The indication is that the gradient is
complex, first rising above the level for the body-standard (carapace
length), then sinking well below it in the merus, then rising again.
It would be of great interest to establish this by obtaining statistically
adequate data, as this is the only indication so far obtained of a com-
plex growth-gradient with two points of inflexion within a single limb.
88
PROBLEMS OF RELATIVE GROWTH
sheep and giraffe which together with inspection of skeletons
makes it fairly clear that in the phylogenetic elongation of the
giraffe's leg, the growth-centre has lain in the cannon-bone,
with a steep gradient distally, a less steep one proximally.
Meanwhile actual weight (and length) measurements made by
Hammond (1927, 1929) and analysed by Huxley (1931B) on
the individual growth of the hind-limbs in sheep, show that
in regard to the pelvis and the three segments femur, tibia
and cannon-bone (unfortunately the digits were not measured)
there is, correlated with the negative heterogony of the whole
limb relative to the body, a reversed
growth-gradient with low point dis-
tally (Table VIII ; Fig. 49). Though
these constitute but two isolated bits
of evidence, they indicate, so far as
they go, that the growth-mechanisms
underlying all heterogony are similar,
and that when heterogony is negative,
the sign of the gradient is simply
reversed.
Hammond (1928, see also 1921)
has also shown that the growth-
gradients in the limbs and elsewhere
affect the muscles as well as the bones,
so that the study is of practical as
well as theoretical importance. An
important point made by Hammond
may be given in his own words.
Sheep
Fig. 48. — Comparison of the
skeleton of the foot in Ox,
Sheep and Giraffe, to show-
graded alteration in propor-
tions of parts.
To effect the transformation form
a typical (e.g. ox) form to that in
the giraffe, y — c has been enlarged,
c — b has remained nearly constant,
b — a has been decreased, and a — o
markedly decreased. In addition,
the length : width ratio has been
increased.
"As the animal grows, it changes its
conformation ; at birth the calf or lamb
is all head and legs, its body is short and
shallow, and the buttocks and loin are
comparatively underdeveloped ; but, as it
grows, the latter — buttocks, loin, etc. — grow at a faster rate than the
head and legs, and so the proportions of the animal change. . . .
The extent to which these proportions change determines its con-
formation ; those which develop most for their age have the best meat
conformation, while those which develop least have the worst . . .
Breed improvement for meat, therefore, means pushing a stage fur-
ther the natural change of proportions as the animal matures. . . .
The adult wild Mouflon ewe is in its proportions but little in advance
of the improved Suffolk lamb at birth, although it is much larger.
What this means to the butcher and consumer is that of 100 lbs.
live weight of an animal shpaed like the Suffolk lamb four days old
the butcher can hang up as carcase in his shop 53 lbs., and the cus-
GROWTH-GRADIENTS IN NEGATIVE HETEROGONY 89
tomer can eat as flesh only 30 lbs. ; on the other hand, when the
animal is shaped like that of the adult Suffolk ram, from 100 lbs. live
weight 67 lbs. of carcase is obtained, and of this 61 lbs. is flesh which
can be eaten — more than double that from of the badly shaped animal."
Phase 1
(Smallest to largest
new-born
Phase 2
(Largest newborn
to medium sized
-J*
o
-i
S-j
o
txo
08
0-6
C4
12 3*3 12345
Axis of limb : distal — >
Fig. 49. — Reversed growth-gradient in organs showing negative heterogony
(limbs of sheep).
The figures on the abscissa represent : i, limb-girdle ; 2, humerus, or femur ; 3, radius + ulna, or
tibio-fibula ; 4, carpals, or tarsals ; 5, metacarpals, or metatarsals, (solid line, forelimb ; dotted line,
hind-limb). The ordinates denote growth-coefficients (k), those below i-o signifying negative hetero-
gony : they are taken relative to vertebral column weight. Size-phase 1 includes smallest to largest
new-born specimens (100 to 256 g. vertebral column weight) ; size-phase 2, largest new-born to half-
grown (256 to 690 g. vertebral column weight). The growth-gradient is at first flat, with slight
positive heterogony ; then steeply tilted downwards distally, upwards proximally.
TABLE VIII
Relative weights of parts of the skeleton of Suffolk sheep at three
different ages (c? Hammond, 1929; $ Hammond, 1927), reduced to
proportions of weight of cannon-bone taken as 100. From Huxley,
1931B.
</l
Vertebrae
c •
° tao
1 v
§1
Us
3
6
IS
In
286
U
s
a
Is
a*—
a
a
~3
0
3
cu
3
X
"3
O,
nj
0
T3
C
a
~5
0
">
u
0
'0
«
u
0
a t-
-s a
"3
S3
■2 •>
u
100
212
O
^4
O
H
J
Birth . .
220
IOO
187
150
75
1097
404
489
330
1230
722
6" 5 months
100
352
320
420
IOO
213
241
169
1 1 79
58l
564
633
1778
1205
i 4 years .
100
380
361
619
IOO
276
280
257
1235
854
775
794
2423
1706
Ratio 4 vrs.:
I -00
1 -3°
i- 70
2-8i
—
i-47
I-Q3
V43
IIS
2-II
1 -.58
2-34
i-97
236
birth
Birth
IOO
147
217
142
9 5 months
IOO
245
285
430
? 4 years .
IOO
272
324
'Sfc'Q
Ratio 4 vrs.:
birth
I -00
1-38
1-50
4-07
go PROBLEMS OF RELATIVE GROWTH
Thus it would appear that one of the chief advances made
by man in creating improved breeds of sheep and other meat
animals has been simply to steepen growth-gradients which
already operate during post-natal development in the wild
ancestral forms. Hammond himself (1927) has expressed a
similar idea. ' The improver of meat-producing animals has
apparently not chosen mutations occurring in isolated points
independently, but rather has based his selection on the
generalized correlated changes of growth '. (See also Fig. 96,
p. 223.)
In consequence of the gradient, there will be much less
difference in the size of the metatarsal between a semi-wild
and an improved breed than in the size of the femur, This
is well brought out by Hammond (1927) in his Fig. 4.
In this connexion, it is well to remember that during em-
bryonic life, the limbs of sheep must show a growth-gradient
precisely opposite in sign to that of their post-natal period.
Lambs are born with relatively long legs, as an adaptation
to accompanying their dams almost from birth. To achieve
these unusual proportions, the leg must have exhibited posi-
tive heterogony during foetal life ; and to allow for the fact
of the later centre of negative heterogony in the cannon-bone,
this same region must have been the positive growth-centre
in the earlier period. The same reasoning applies to Ocypoda,
whose young are similarly precocial.
§ 4. The Form of Growth-gradients
Analysis of the data of Kemp and his co-workers on Palae-
mon spp. undertaken by Miss I. Dean (unpublished) gives
a further interesting result. In these prawns, both male and
female have obviously heterogonic chelae, but the male's
heterogony is considerably higher. Thus a male and a female
of the same absolute size will possess chelae of very different
sizes, the female's being considerably the smaller. But if we
take a male chela and a female chela of the same absolute
size (which will of course be borne by a small male and a
large female body) the proportions of the separate joints will
be found to be fairly similar. This indicates that whenever
marked heterogony, or at any rate heterogony designed to
give rise to a large chela, is present, it must operate by essen-
tially the same growth-mechanism within the limb (and a
mechanism quite different from that in a slightly heterogonic
pereiopod), whether the growth-coefficient of the whole limb
THE FORM OF GROWTH-GRADIENTS 91
relative to the body be moderate or high. The growth-
gradient of the female is not quite so steep as that of the
male, a fact also brought out by Tazelaar on P. carcinus
(p. 92) ; but the male and female chela-gradients are much more
like each other than they are to the gradients of any of the
pereiopods.
Still further proof of the radical difference of the growth-
gradients leading to pereiopod and to large chela is afforded by
the male hermit-crab Eupagurus (Bush, 1930 ; Bush and
Huxley, 1930). Here the right chela during early life is not
much enlarged, and its growth-coefficient is no greater than
that of the pereiopods ; only later does it begin the marked
1,6
1,5
1,4
1*.
1,0
I TTL C
dJstaJ -
-■ 1
1
1 1
— B
-
-
-B^<7
-
/
*^
•
^
/
/
Ax'
■
1
1 '
-xA
1
Fig. 50. — Change in form of growth-gradient with increase of growth-rate in
large (right) male claw of the hermit-crab, Eupagurus.
i, ischium ; m, merus ; c, carpus ; p. propus ; d, dactylus. A — A, juvenile phase ; the growth-
gradient resembles that of a pereiopod. B — B, phase of heterogony of right chela ; the main growth-
centre shifts distally.
heterogony which provides its definitive enlargement. And
during the earlier period its growth-gradient is similar to that
of a pereiopod, with centre in the merus ; while so soon as
the final heterogony becomes marked, the main growth-centre
shifts to the propus (Fig. 50).
Tazelaar (unpublished) has also collected facts bearing on
this subject. In Palaemon carcinus, there is a change in the
growth-coefficient of the chela in both sexes at about 4-5 cm.
carapace length. In the female, before this, the chela has
been growing less rapidly than the neighbouring pereiopods ;
after this it exhibits a considerable heterogony. During the
first of these phases its growth-gradient is almost flat, like
those of the pereiopods, but with a slight growth-centre in
92 PROBLEMS OF RELATIVE GROWTH
the propus. Later it exhibits a marked growth-gradient with
centre in the dactylus.
In the male, the chela in the first phase shows definite
heterogony, about the same as the female chela in the second
phase. During this phase it shows a growth-gradient with
isch. merus carpus prop dact isch. merus carpus prop dact. isch merus carpus prop, dact
1st. pereiopod Cheliped (2nd pereiopod) Cheliped( 2nd. pereiopod)
Fig. 51. — Growth-gradients in the 1st pereiopod, and the chela of the prawn,
Palaemon carcinus.
, male ; , female, (a) 1st pereiopod ; the gradient is flat and close to unity throughout ;
(b) and (c) 2nd pereiopod (chela). (6) 1st phase ; female with slight initiation of growth-centre
distally ; male with regular growth-gradient (distal growth-centre), (a) 2nd phase; female with
definite growth-gradient but incomplete proximally ; male with very marked growth-gradient (sub-
termunal growth-centre).
centre in the dactylus. During the second phase, the male chela
shows extremely marked heterogony ; and it now possesses a
striking growth-gradient, with centre in the propus (Fig. 51).
It would seem as if the steepening of the gradient began near
the top, and then gradually extended centripetally (cf. p. 168).
§ 5. Growth-gradients in Regions of the Body
Precisely similar gradients to these found in appendages
may be traced in the growth of whole regions of the body.
REGIONAL GROWTH
93
The most clear-cut examples concern the abdomen of crabs,
which in all cases are narrow in the male, broadly expanded
in the female. A large series of measurements has been made
by Sasaki (1928) on both sexes of the Japanese species Tel-
messus cheiragonus. Analysis of these data shows that whereas
the growth-gradient for breadth in the male abdomen is nearly
2.5
©
id
4)
a
"aS-d 2.0
s —
.2 a.
4J
1.3
o
u
O
3 4
Segments of abdomen: distal — >
Fig. 52. — Growth-gradients in the abdomen of crabs.
Solid lines, for breadth of abdominal segments: ©, Telmessus cheiragonus, <$ ; X, Telmessus
cheiragonus, 9 : +, Pinnotheres pisum, $. Dotted line, for length of abdominal segments in
Pinnotheres pisum, ? .
flat, with its growth-centre, if so it may be called, near the
centre of the region, in the female it is steeper, with its centre (as
in the typical male chela) in the penultimate segment (Fig. 52).
These figures may be compared with those cited for the
edible crab, Cancer pagurus, by Pearson (1908, p. 21) in two
large specimens of the same size but opposite sex (Table IX).
TABLE IX
Cancer pagurus; from data of Pearson, 1908.
Carapace-breadth 235 mm.
9/6" ratic
per cent.
Abdomen
1
segment
Length
Breadth
Length
Breadth
Length
Breadth
mm.
mm.
mm.
mm .
I
17
22
17
25
IOO
114
2
8
17
8
22
100
129
3
7
23
7
30
IOO
130
4
8
20
8
32
IOO
160
5
9
17
10
35
III
206
6
13
16
20
35
154
219
7
13
13
18
21
138
162
94
PROBLEMS OF RELATIVE GROWTH
I have (last column) calculated the ratios of ? to £.
These indicate clearly that both for length and breadth there
exists a growth-gradient in the ? abdomen, with high point
in the sixth segment, though the two gradients must be of
very different
slope and shape.
Precisely simi-
lar results have
been obtained for
the female ab-
domen of the
common Carci-
nus maenas (un-
published). That
the growth-
centre need not
be in the pen-
ultimate seg-
ment, however, is
shown by the
Pea-crab, Pin-
notheres pisum.
Scale drawings of
this at different
stages of its
growth are given
by Atkins (1926) ;
I have measured
these, and al-
though the num-
ber of specimens
is very small, it
is quite enough
to demonstrate
the gradient. The
gradient is first
of all interesting
because of its steepness. The growth-coefficients of the separ-
ate segments, relative to carapace length, range up to 2-3
and over, a very high figure. In passing, this remarkable
heterogony is doubtless correlated with the small absolute size
to which the crab is restricted within its host's shell ; the
animal has to attain its full female proportions at a much
Fig. 53. — Changes of shape in the female abdomen of
the pea-crab, Pinnotheres, during growth.
Above, stage I, late (carapace width about 3 mm.) ; centre, stage
II 16 (carapace width 6 mm.) ; below, final form, stage V (carapace
width 1 1 -5 mm.). Thp abdomen is at first male-type, but shows
marked heterogony, with terminal growth-centre, until it overtakes
the bases of the legs.
INTENSITY IN DIFFERENT PLANES OF SPACE 95
smaller absolute size than is necessary in most crabs. (Fig.
52.) (See Huxley, 1931B.)
During growth, the young female type of abdomen, which
with its flat or concave margins resembles the male's, is con-
verted into an almost circular structure, with the fourth and
fifth segments the broadest. Casual inspection would indicate
that one of these segments must contain the growth-centre ;
but casual inspection is wrong — the terminal segment (telson)
is so small in the young female that for it to achieve its only
moderate definitive size it must, and does, contain the growth-
centre (Figs. 52, 53).
In the heterogenic growth of the face of mammals, which
we have noted in Chapter I, it would appear from the work
of Todd (1926) that in the palatal region the centre of maximum
growth lies in the palatal processes of the maxillae ; there
is a moderate amount of growth in the premaxillae and very
little in the palatals. It is impossible to arrive at quantitative
expression, but the facts are consonant with the idea of a
double gradient culminating within the maxillae.
§ 6. Graded Growth-intensity in the Different Planes
of Space
So far, we have been considering growth-gradients referring
either to weight, or to linear measurements in one axis.
In Pinnotheres, measurements were taken both for the
length and breadth of the abdominal segments (Fig. 53) ;
when we consider the two sets of measurements in relation
to each other, we find that the ratio between growth-coefficient
for breadth and that for length is as follows for the distal
half of the female abdomen :
4
5
6
7
. growth-coefficient for breadth
' growth-coefficient for length
1-62
1-46
1-40
1-36
Thus the difference between male-type and mature female-
type abdomen is brought about (1) by greater growth in the
female, both in length and in breadth ; (2) by the breadth-
growth being throughout higher than the length-growth ;
(3) by the excess of breadth-growth over length-growth being
96
PROBLEMS OF RELATIVE GROWTH
greatest, in the segments measured, in the fourth segment,
and decreasing distally. Without further measurements it is
impossible to state whether the high point in breadth-growth
predominance is really in the fourth segment, or further
basally ; but at least the orderly and graded relation between
the intensity of growth in the two planes of space is clearly
brought out.
But a more complex, and perhaps more interesting problem
is afforded by the two chelae of markedly heterochelous Crus-
tacea of which the lobster (Homarus) is the most familiar,
and the pistol-crab (Alpheus) the most extreme example. In
all of these, the heavier (crusher) claw is broader and alto-
gether bulkier in build ; and the lighter (nipper) claw always
has a relatively and often an absolutely longer dactylus
(Huxley, unpublished : see figures in Przibram, 1930). It
would seem to mere inspection that the fundamental difference
between the two claws lies in the difference of their growth-
gradients in the three planes of space, the growth-coefficients
for breadth and depth being relatively as well as absolutely
much higher in the crusher, and the growth-centre as regards
length being more distal in the nipper.
A few measurements have been made on the chelae of
lobster (Homarus) to check this (Huxley, unpublished). So
far as they go, they confirm the impression made by inspec-
tion. The (only very approximate) value of the growth-
coefficients are as follows (relative to carpus length as standard) :
TABLE X
Approximate Growth-coefficients (k), relative to carpus-length,
of the linear dimensions of the segments of the chelae of
the European Lobster
Crusher
Nipper
Length
Breadth
Depth
Length
Breadth
Depth
Dactylus
Propus .
Carpus .
Merus
Ischius .
0-6
o-95
(i-o)
0-85
0-85
1-2
1-2
o-95
09
o-85
I-i
I-I
II
I-o
0-9
o-85
o-8
(i-o)
0-85
o-8
1-05
1-05
1-05
115
I 05
i-o
0-9
i-i
1-05
It will be seen that the chief difference between the two
claws is in the distal region. In the crusher dactylus, the
INTENSITY IN DIFFERENT PLANES OF SPACE 97
length-growth falls off notably to a much greater extent than
it does in the nipper dactylus. In the nipper, breadth-growth
is approximately constant in all joints, while depth-growth
decreases distally. In the crusher, on the other hand, depth-
growth increases steadily, and breadth-growth markedly, from
ischium to propus, to fall slightly in the dactylus ; (it should
be recalled that the rates of growth are taken relative to the
claw's own carpus-length, not to an independent standard in
the body). Perhaps the most interesting point is the change
20
Homarus:
crusher
10
0 8
nipp
en
14-
10
m.
c
distal
m
c
distal
B
Fig. 54. — Probable change in the ratio between growth-intensity in depth
and that in length (solid line) and the ratio between growth-intensity in
breadth and that in length (dotted line) along the axis of (A) the crusher-claw
and (B) the nipper claw (right) of the Lobster.
i, ischium ; m, merus ; c, carpus ; p, propus ; d, dactylus.
in the ratios between the growth-intensities in breadth and
depth to that in length, as we pass along the axis of the limbs.
The present figures are based only on a few specimens, and
curves constructed directly from them are rather irregular.
I have therefore ventured to prepare smoothed curves which
represent to my mind the most probable state of affairs
(Fig- 54)-
Thus the main difference between crusher and nipper is not
merely one of total growth-potential, but also of the different
7
98
PROBLEMS OF RELATIVE GROWTH
distribution of growth-potential in the three planes of space.
The minor differences, e.g. number and form of ' teeth ' are
presumably due to specific gene-differences, not correlated
directly with growth.
A further point of interest was elicited in these chelae by
making measurements not
only of whole segments, but
to intermediate points marked
by spines, etc. (notably the
four large spines on the median
side of the propus). These
measurements indicate that,
as suspected, the growth-co-
efficient of a given whole seg-
ment merely represents a
mean value, the growth-
gradient being real and con-
tinuous, and that the values
of the growth-coefficients are
altering continuously and
regularly along the segment.
Many more measurements,
however, would be necessary
before the true gradients could
be plotted in detail, and to
obtain these would be a very
difficult task, as fixed points
to measure are not numerous
enough. (See also Locket, p.
261.)
Tucker (1930) has made
measurements on the chela of
the anomuran Upogebia lit-
toralis, for normal specimens
of both sexes as well as for
those parasitically castrated
by the Bopyrid Gyge branch-
ialis. The length measure-
ment chosen was (propus -f dactylus) while the breadth measure-
ment was propus-breadth. The results for normal specimens are
shown in Fig. 55. For both males and females, two distinct
phases of growth are found, the alteration occurring at 12-13
mm. carapace length. In the first phase of both sexes there is
6 8 10 13 16 19
carapace length, mm.
Fig. 55. — Upogebia littoralis (Deca-
poda, Reptantia, Anomara). Chela
(dactylus + propus) length and chela
(propus) breadth against carapace
length ; logarithmic plotting. Ste
Table XI.
X , males ; 0, females; .parasitized
males ; , parasitized females. The male
chela, as in Uca, begins by being distinctly
heterogonic and then becomes less heterogenic.
For details, see text.
(From data of Tucker, 1930.)
INTENSITY IN DIFFERENT PLANES OF SPACE 99
a good approximation to constant differential growth-ratio ;
this exists also in the males' second phase, with a slight
falling off in the last point. The male chela shows positive
heterogony in both phases. In the female, there is a pro-
gressive falling off in the growth-coefficient during the second
phase, more marked for breadth than for length. The female
chela is positively heterogonic in the first, negatively hetero-
genic in the second phase. If the growth-coefficients are
calculated for the whole of the first phase, and between the
first and third points of the second phase, we obtain the
following result :
TABLE XI
Growth-coefficients for Length and Breadth of Chela in
Upogebia littoralis, calculated from the data of Tucker,
1930
k for
chela breadth
k for
chela length
_ .. r k breadth
Ratio of 7— ; tt—
k length
Males, first phase
Males, second phase .
1-82
1-47
1-37
I-I3
i-33
1-30
Females, first phase .
Females, second phase .
i-37
0-72
1-05
0-90
1-30
o-8o
I.e. the ratio of growth-intensity for breadth to that for length
remains the same within the limits of experimental error so
long as the chela is positively heterogonic. When, however,
as in the females' second phase, it becomes negatively hetero-
gonic, the situation is reversed, and growth-intensity for
breadth falls below that for length. This reversal would
appear to be analogous to the reversal of sign of the growth-
gradients in negatively heterogonic organs (p. 87). Logarith-
mic plots (see Fig. 55) as well as Tucker's graphs indicate
some interesting points with regard to the action of the para-
site upon chela growth. The parasite has a general feminizing
action, very similar to that of Sacculina. In males, the effect
in reducing the growth-coefficient of chela-length is slight up
to 10-5 mm. carapace-length, for chela-breadth up to 8 mm.
carapace-length. From then on the growth-coefficient for
length remains just above that for females throughout ; that
for breadth, after approximating closely to that for females
until the second phase, fails to be as much reduced as the
ioo PROBLEMS OF RELATIVE GROWTH
females during the remainder of life (k = about 0-95 instead
of 072).
In the parasitized females, k for length is throughout just
below that of normal specimens, whereas k for breadth descends
well below the normals between 10-5 mm. and 13-5 mm.
carapace-length, then gradually ascending almost to the level
of the normals. This latter fact would seem to indicate that
in females the parasite (as in Sacculina) acts as would a pre-
cociously functioning ovary, but that the value for normal
chela-breadth in the larger normal specimens represents a final
partition-coefficient towards which parasitized as well as nor-
mals tend to approach. It is further clear that breadth-growth
is more sensitive than length-growth to ovarian influence and
to the parasite's pseudo-ovarian influence, and reacts to them
both earlier and to a greater degree. The failure of the chela
of parasitized males to respond in breadth-growth to the
influence of the parasite as much as the normal female chelae
to the influence of the ovary may be due to a specific difference
in male and female chela-tissue, but it is more probably due
to the fact that the degree of feminization effected by the
parasite is variable. As breadth-growth is more sensitive than
length-growth to feminizing influences, this variability will be
reflected more markedly in the curves for breadth-growth.
Finally, the work of Hecht (1916) on the growth of Teleost
fish is of some interest here. He finds in a number of species
of markedly differing body-form that the relation of the maxi-
mum width of the body to total length remains fairly constant
(to about ± 16 per cent.) whereas the relation for depth is
highly variable (by over ± 55 per cent.). The form of fish
appears thus to be determined much more by changes in the
growth-coefficient for depth than by changes in that for width.
§ 7. Gradients in Growth-rate of Epidermal
Structures
Another extremely interesting case of growth-gradients is
provided by the work of Juhn, Faulkner and Gustavsen (1931).
Working on the domestic fowl, they find in the male (normal
and castrated) definite regional gradients in the rate of growth
of feathers which are regenerating after plucking. The most
marked of these regional gradients is in the breast, and has
its high point posteriorly (Fig. 56).
In addition, there is in normal and castrated males a
dorso-ventral gradient in breast, back, and saddle, with high
EPIDERMAL STRUCTURES
101
point dorsally, e.g. the regenerating saddle feathers of capons
grow more slowly than the breast feathers. A similar gradient
exists in the breast, but not in the saddle.
Two further points demand special attention : one is a quite
new one, of obviously first-class physiological importance —
viz. that in all feathers low rate of growth is associated with
a low threshold of sensitivity to the sex-hormone, as our
authors have demonstrated by means of an ingenious series
of experiments. Accordingly, saddle feathers respond by
Fig. 56. — Gradients in feather-growth in fowls ; constructed from the data
of Juhn, Faulkner and Gustavsen, 1931. The ordinates represent rates of
feather-growth per day, in mm.
In A, the abscissa axis represents the anteroposterior axis of the breast region in a capon (0, points
for single feathers ; ©, means). In B and C, it represents the gradient downwards from the mid-
ventral line (in B, in both anterior and posterior breast regions of a cock ; in C in the anterior breast
region of a capon). In D, the rates of growth of anterior and posterior regions of the breast of a
cock and a hen are compared.
feminization to small doses of female hormone which have
no effect upon breast feathers : or again, a single injection
of female hormone capable of inducing a large feminized
pattern on the slowly regenerating feathers of the anterior
breast produce smaller bars of feminized pattern on the more
rapidly-growing feathers of the posterior breast. (And see
p. 260.)
The other point is a confirmation from this quite new
quarter of the principle we found to hold in crustacean
102 PROBLEMS OF RELATIVE GROWTH
limbs, etc., that increase in relative rate of growth in an
appendage or region was accompanied by a steepening of the
growth-gradient. This holds even for the type of growth-
gradient affecting feather-growth only, which we are here
considering. In females and feminized males, the absolute
rate of growth of, e.g., the breast-feathers is reduced, and the
feathers, growth-gradient in this region is at the same time
almost flattened out (Fig. 56D).
Mr. Miller, of the Animal Breeding Research Department,
Edinburgh, has pointed out to me that similar gradients
affecting the growth of epidermal structures occur in mammals.
For instance, in (untrimmed) manes of horses, the length of
the hairs increases steadily from the ears to the middle of the
mane, then sinks steadily to the hind end of the mane. A similar
gradient in hair-length occurs in the human beard, with high
point medially. In such cases we are only dealing with total
amount of growth as given by the definitive length of the
hair : I do not think anything is known about the rate of
growth or of regeneration. In the fowl, Juhn and her collabora-
tors (1. c.) find the following interesting facts. In breast, and
also in the back-and-saddle region, there exists a distinct
antero-posterior gradient in definitive feather-length. In the
breast, this is correlated with the above-mentioned gradient
in regenerative growth-rate, but also with the fact that
regeneration continues for a longer time in the posterior
feathers. Presumably the same two factors are at work in
normal growth. In back and saddle (information in a letter
from Dr. Juhn) there is no gradient in regeneration-rate, but
only a gradient in the length of time for which regeneration
proceeds. We thus have two distinctive methods of growth,
both capable of gradation, affecting the size of epidermal
structures.
§ 8. Conclusion
The chief points brought out in this chapter are the follow-
ing. Differential growth of a limb or appendage or a well-
marked region of the body appears never to be brought about
by an equal distribution of excess growth-potential throughout
the organ or region. On the contrary, the growth-potential
of the organ or region is distributed in the form of a growth-
gradient, normally with a single high point or growth-centre,
from which growth-intensity grades downwards in both direc-
tions (or in one, if the growth-centre be terminal). In general,
CONCLUSION 103
the less the difference between the growth-coefficient of the
organ or region and that of the rest of the body, the less
marked and flatter is the gradient, so that in isogonic organs
there is scarcely any gradient, but all the parts grow at approxi-
mately the same rate as the body as a whole. There are,
however, some unusual gradients, as in the first antennae of
certain copepoda, in which the growth-coefficients of part of
the organ are above, the rest below unity.
When one of two corresponding organs is positively, the
other negatively, heterogonic, the growth-gradients of the two
appear to be similar but reversed in sign, the same joint
being in one a centre of maximum, in the other a centre of
minimum growth.
In most markedly heterogonic organs so far investigated
(chelae of Crustacea, abdomens of female crabs, limbs of
ruminants) the growth-centre is terminal or sub-terminal.
When an organ is markedly heterogonic in both sexes, then,
even though the growth-coefficient of the whole organ may
differ considerably in male and female, the growth-gradient
within the organ appears to be essentially similar — i.e. there
is a certain qualitative type of growth-gradient needed to
produce an organ of a certain morphological type, though
quantitative details may be different. This is confirmed by
the change from a growth-gradient with sub-basal centre (as
in the pereiopods) to one with a sub-terminal centre (as in other
large chelae), when the chela of the male Eupagurus passes
from a slight heterogony, no greater than in the walking legs,
to the marked heterogony of maturity.
The gradients for growth in the three planes of space may
be different ; applications of this are seen in the abdomen
of male and female crabs, and still more in the crusher and
nipper claws of heterochelous Crustacea.
The existence of growth-centres and growth-gradients is an
empirical fact, whose physiological explanation is quite un-
known but may prove to be of importance for the study of
growth in general.
CHAPTER IV
GROWTH-GRADIENTS AND THE GENERAL
DISTRIBUTION OF GROWTH-POTENTIAL
IN THE ANIMAL BODY
§ i. General Growth-gradients : D'Arcy Thompson's
Graphic Method
THE next step is to inquire whether the growth-
gradient mechanism may not underlie the general
growth of the body as well as the growth of specialized
appendages or regions. In other words, we want to see whether
the sharply-marked growth-gradients of a chela or an abdomen
are not merely special cases of a more general but still orderly
mechanism underlying the distribution of what we may, for
want of a better term, call growth-potential, throughout the
whole organism.
A strong indication that this is so was afforded by D'Arcy
Thompson (1. c, chap. 17) through his ingenious application of
the principle of Cartesian co-ordinates to the problem of
animal form. Let us take the most spectacular and at the
same time one of the simplest of his instances. The strange
fantastic sun-fish, Orthagoriscus, is a close relative of such
types as Diodon ; but its adaptation to an almost planktonic
life at the surface of the sea has led to a change of form so
radical that it is at first sight difficult to see how it could have
been brought about within any short space of evolutionary
time. D'Arcy Thompson, however, pointed out that if you
inscribed the outline of a Diodon in a framework of rectangular
co-ordinates, and then distorted this in a certain perfectly
regular way, you would obtain a very close approximation of
the outline of an Orthagoriscus (Fig. 57). From the figure
it will be immediately obvious that the essence of the trans-
formation, considered biologically and not merely as an
exercise in higher geometry, must have been the origin of a
very active growth-centre in the whole of the hind-region of
the body, whence the intensity of growth diminished regularly
104
GENERAL GROWTH-GRADIENTS
105
towards the front end. In other words, superposed on what-
ever growth-mechanisms may be necessary to generate a form
similar to that of Diodon, there has arisen a steep and unitary
postero-anterior growth-gradient extending throughout the
entire body, with high point almost or quite at the extreme
hind end.
The diagram, however, also illustrates the serious limitations
of the method. In the first place,
two adult forms are contrasted. ,e
What should rather be contrasted
are, phylogenetically, the young
form of the presumed Diodon-like A \d
ancestor, and the adult Ortha-
goriscus ; or, ontogenetically, a
young Diodon-like stage of Ortha-
goriscus with the adult condition.
Fig. 57. — Cartesian transformation of the
outline of the teleost fish Diodon (left) to
give the outline of the sun-fish, Ortha-
goriscus (right).
But this is not so serious as the following point : that even
should you compare the correct two stages, this graphic
method, if interpreted in terms of growth, can only give a
general and qualitative picture of the mechanism at work,
in place of a specific and quantitative one. For if, as D'Arcy
Thompson points out, the transformation, so difficult to under-
stand at first sight, becomes readily comprehensible on the
idea of an orderly change in the distribution of growth-activity
106 PROBLEMS OF RELATIVE GROWTH
along the axis of the body, then clearly the proportions of the
animal must be continually changing so long as it is increas-
ing in absolute size, or at least over a long space of time.
But the fish's outline and the system of co-ordinates drawn
to fit it, represent the state of affairs only at one particular
moment of its life-history. If the fish had grown to twice
the bulk, its proportions would have changed, and the co-
ordinate grid would have to be altered ; yet the underlying
growth-gradient might have remained wholly unaltered.
An improvement would be effected if the absolute sizes of
the two outlines which mark the onset and stoppage of orderly
differential growth could be given : but even so, the real
invariable, namely the growth-gradient expressing the values
of the growth-coefficients along the body, could only be
obtained from this by calculation, and is not deducible by
inspection.
For this reason, the co-ordinate method, while of the utmost
importance as affording a graphic and immediate proof of
the need for postulating regularities in the distribution of
growth throughout the body, is of little use for detailed analysis,
because by its nature it neglects the fundamental attribute
of differential growth, namely the change of relative propor-
tions with absolute size : it is static instead of dynamic, and
substitutes the short cut of a geometrical solution for the more
complex realities actually underlying biological transformation.
None the less, it is invaluable as demonstrating the need
for thinking in terms of growth-gradients and, in general, of
an orderly system in the distribution of growth-activity
throughout the body : and this whether we are considering
individual or evolutionary change of form. We have only to
glance at D'Arcy Thompson's figures of brachyuran carapaces,
ungulate limbs, and so forth, to realize immediately its utility
in this respect (Figs. 48, 58).
Another interesting use has been to deduce the course of
evolution over gaps where actual fossil data are missing. It
is easy to say that we can do this by common-sense, simply
inserting hypothetical intermediate stages between known
end-points. But this is insufficient. There are, for instance,
hundreds of possible ways of bridging the gap between the
pelves of Archaeopteryx and the cretaceous bird Apatornis,
to take an example worked out by Thompson ; but if the
evolutionary modification of such a structure be due to growth-
changes, and if orderly and regular growth-gradients be the
GENERAL GROWTH-GRADIENTS
107
mechanism by which growth-changes must operate, then you
can deduce the precise course of evolution within comparatively
narrow limits (see Fig. 59).
As showing the validity of the method, D'Arcy Thompson
has applied it to the evolution of the skeleton of the horse,
Fig. 58. — Carapaces of various crabs to show how they may be readily derived
from one form by simple Cartesian transformation.
and then compared his deduced results with the known fossil
forms (Fig. 60).
The agreement of actual with deduced skull-form is very
close for Protohippus, Miohippus and Mesohippus. But the
species of Parahippus figured by Thompson, as he points out,
r 3 "»
6 7 8
— |
^
)ft
^
j>
1 J L
* -.
/v
y
s
/
1
1Xi*Si769
A
Fig. 59. — B, C, and D, Hypothetical intermediate stages, constructed by-
applying the method of Cartesian transformation, between A, pelvis of
Archeopteryx and E, that of Apatornis.
108
: \ \ LA....
7"! "i t 5 * T 9 ■) '0 "
Fig. 6o. — Comparison of hypothetical forms of the skull in the ancestry of
the horse (C — E), constructed by applying the method of Cartesian trans-
formation (A — H) with actual forms ; (M) Mesohippus ; (Mi) Miohippus ;
(Pa) Parahippus ; (P) Protohippus.
109
no PROBLEMS OF RELATIVE GROWTH
has a straighter skull, with higher nasal region, than demanded
by theory : and he therefore suggests that this form is not on
the direct line of equine descent. The genus Parahippus is
generally recognized as ancestral to Equus (see Matthew, 1926),
but it contains a number of ' widely varying distinct species '
(Matthew, 1. c, p. 160). We may therefore agree with Thomp-
son that the particular species he has figured is an aberrant
type. This is a remarkable achievement, and clearly provides
a new method in detailed paleontological research, which is
clearly of real value.
If, however, a way could be found of taking account of the
changes in absolute size which so frequently accompany direct-
ive evolution, and automatically induce changes in proportions
of limbs, etc., a further analysis might be possible, which would
enable us to distinguish what we might call the consequential
changes in form from the strictly adaptive, meaning by the
former those changes in proportions which, unless counter-
acting growth-mechanisms are evolved, automatically accom-
pany change of size, and by the latter such changes as are
specifically related to mode of life, and presumably are brought
about by natural selection (which, of course, in such case must
act via the genes which control the growth-gradients). But
of these and other evolutionary bearings of the existence of
growth-centres and growth-gradients I shall deal more fully
in a later chapter.
The graphic method of D'Arcy Thompson enables us to
postulate that orderly growth-gradients must exist in the
body as a whole, though from his figures it is further clear
that the main system of gradients need by no means be so
simple as in the case of Orthagoriscus, but that a number of
gradients may be combined even along a single axis of the
body ; further, that minor gradients may be locally super-
posed upon major ones ; and that the various components of
the system as a whole appear to interact with and modify
each other.
§ 2. General Growth-gradients : Quantitative Analysis
Our further task is to try to find out whether these growth-
gradients can be expressed, even approximately, in quanti-
tative terms, and whether we can discover any empirical
rules concerning the nature of the influence exerted by one
gradient upon another. To do this is no easy task, for it
demands quantitative measurements, either of mass or linear
QUANTITATIVE ANALYSIS in
dimensions, of numerous parts of the body ; and these must,
for one thing, be made at a considerable number of absolute
sizes, to make sure that the gradients, etc., do not change
with age, and for another be made on a considerable
number of specimens within each size-group, to exclude the
effects of random sampling. Such a body of measurements
only exists for very few animals, and even there not for
sufficient points or organs to give a complete picture of
the growth-gradients ; but in spite of these limitations, the
results are of considerable interest. We will begin with
the case of the hermit-crab, Eupagurus, analysed under
my direction by S. F. Bush (Bush, 1930 ; Bush and
Huxley, 1930). Measurements were made of the eyestalks,
first and second antennae, third maxillipeds, chelae (= first
pereiopods), pereiopods 2-5, and uropods, as well as of median
pre-thorax length as standard ; (in some appendages the
length of an arbitrary portion was taken in place of total
length). When groups of small and large body-size were
compared, the following significant facts emerged. We will
first consider males, and in them the left side of the body,
where matters are not complicated by the pronounced asym-
metric heterogony of the right chela. Here we find a quite
definite double growth-gradient, with growth-centre in the
third pereiopod, whose growth-coefficient is considerably higher
than that of the body (pre-thorax) ; from this point growth-
activity falls off both anteriorly and posteriorly, until it is
about equal to that of the body in the third maxillipeds and
fourth pereiopods respectively, while less in the cephalic
appendages and uropod (Fig. 61). There is a slight irregularity
in the region of the last pereiopods ; otherwise the gradient
is quite regular.
This is remarkable, since the organs measured are not only
of very different shape, but differ vastly in absolute size ; yet
the short eyestalk, first antenna and third maxilliped, the
long second antenna, the large first to third pereiopods and
the small fourth and fifth pereiopods, all, as regards their
growth-ratios, fall into this regular graded series.1
The further significant fact emerges that although we are
only considering the growth of localized appendicular organs;
1 This appears doubly surprising, since the size of an appendage
must in the long run depend upon its relative growth (growth-ratio) ;
we shall meet with the probable explanation of this paradox later
(§3. P- 118).
112
PROBLEMS OF RELATIVE GROWTH
60
Left Side
w
20
Right Side
W 60
the gradient appears to be a continuous one. In other words,
although appendicular growth can only take place in relatively
small, pre-localized regions, yet the gradient determining the
amount of that growth is continuous. The growth-gradient
is thus probably something of a very fundamental nature,
akin to those gradients (of equally recondite nature) which
Boveri, Child, von Ubisch, Weiss and others have found it
necessary to pos-
tulate in the early
development of the
egg and in regenera-
tion to account for
polarity and certain
orderly phenomena
of morphogenesis ;
the gradients hap-
pen to exist, and
where growth is pos-
sible, they influence
the amount of that
growth. We shall
be forced to draw
similar conclusions
in Chap. V (p. 152).
(And see p. 262.)
However, that
such a general body-
gradient can be
locally modified is
clearly shown by
looking at the curve
for the right-hand
side of the males,
where a second high
point is made by
the chela.
Further proof of the graded regularity of the changes in
growth-intensity is seen when we investigate the asymmetry
of the hermit-crab in the two sexes (Bush's Fig. 9). Males and
females are approximately symmetrical in the head region
(though females appear to be slightly right-handed anteriorly,
slightly left-handed rather more posteriorly). The male be-
comes markedly right-handed in the chela region, and then
Egestatks
Antenna f
Antenna 2
Maxil/iped3
Cheia
PereiopodZ —
Pereiopod3
Pereiopodt
Pereiopod5
Uropod
60 W 20 W 60%
Percentage increase in iengths of Appendages
Fig. 61. — Growth profile of the hermit-crab.
Growth-gradients along the body of male and
female hermit-crabs (Eupagurus prideauxi).
The figures give percentage increases of appendage length for a
given percentage increase of prothorax length (marked as dotted
line).
QUANTITATIVE ANALYSIS 113
progressively more left-handed. The same is true for females,
but the right-handedness is less in the right-handed region,
the left-handedness greater in the left-handed region. In
accordance with this greater right-handedness of the males
(which is doubtless correlated with the extreme right-handed-
ness of the male chela), the male is still slightly right-handed
at the level of the fourth pereiopod, while the female is here
symmetrical ; i.e. the change from right- to left-handedness
is not associated with a particular appendage, but takes place
in relation with a graded distribution of growth-intensity.
An extremely interesting case of a large-scale growth-gradient
extending through much of the body is provided by the stag-
beetles (Lucanidae). It is well known to Coleopterists that
large males are characterized, not only by relatively large
mandibles, as we have already seen, but also by relatively
large heads, prothorax and prothoracic legs, although the
relative increase for these is much less than for the highly
heterogenic mandibles. So far as I am aware, however, no
measurements have been available until those recently made
by Mr. Edwards at my suggestion and analysed by me (Edwards
and Huxley, unpublished). They refer to L. cervus.
The partition-coefficients of various organs, against elytron-
length, in males are as follows (between elytron lengths 15
and 21 mm.).
TABLE XI
A
landible 1
Antenna 1.
Head 1.
Head br. Prothorax 1.
2-35
I.38
1-62
1-59 i-oo
3r. Wing 1.
I-OO
First leg 1.
Prothorax 1
I-I2
Femur Rest of leg
1-09 i-oi
Second leg 1.
Third leg 1.
Femur
I -04
Rest of leg
0-97
Femur Rest of leg
0-95 0-98
There is thus, in general, a decrease in growth-intensity from
front of head to hind limb, in other words, a growth-gradient
with centre in the mandibles. Heterogony changes from
positive to negative between the first and second leg. (It may
be noted in passing that the gradient is a little steeper in the
femora than in rest-of-leg — evidence of a graded effect within
the legs.)
The data for the females are not so full, but indi-
8
H4
PROBLEMS OF RELATIVE GROWTH
cate a gradient of wholly
nearly flat, with very low
head, pro thorax elytron
Fig. 62. — Growth-gradients in the stag-
beetle, Lucanus cervus.
(A) Ratios of the growth-coefficients of the male
to those of the female for various organs and regions.
For head, prothorax and elytron, the ratios are
calculated from the mean between the coefficients
for length-growth and for breadth-growth (B)
Ratios between growth-intensity in breadth and
growth-intensity in length in 3 regions of the body.
different shape, much more
values for the head appen-
dages, but higher values
for the legs, which show
positive heterogony. It is
worth noting that the dis-
tribution of growth-inten-
sity in length and in breadth
differs in the two sexes
(Fig. 62). In both the in-
tensity of breadth-growth
diminishes faster than that
of length-growth as we pass
forward along the body, but
the diminution is greater
in females (cf. p. 98).
The best way of compar-
ing the two sexes is to take
the ratios of the partition-
coefficients of different
organs in males and females,
and plot these (Fig. 62).
It would appear that the
main change involved in
transforming the female to
the male type has been the
formation of a very high
growth-centre in the man-
dible region. With this
are correlated positively
heterogonic effects in the
immediately neighbouring
regions. But further pos-
teriorly, in the leg region,
the gradient is continued
into a phase of negative
heterogony. It would thus
appear that not only may
a growth-gradient be steep-
ened, but that marked
steepening of one end of
a gradient may actually
somewhat depress its other
QUANTITATIVE ANALYSIS
ii5
end : the gradient is not only steepened but tilted round a
point within its length (cf. Hammond's sheep limbs, p. 88). x
Fig. 62 shows the essential facts. It will be seen that besides
the parts mentioned, there is also a slight heterogony of the
male antennae and prothorax (and see Table XIa) ; further,
that in the regions of the body measured, heterogony in
length and breadth is quantitatively different. The most
significant point is that the heterogony of the three pairs of
legs is graded in an antero-posterior direction. The facts in
general support the conception that there exists a growth-
gradient with centre in the mandibles, grading down pos-
teriorly 2 (Fig. 63).
Theoretically, two further points appear to me particularly
interesting. The first is this,
that we can exclude functional
hypertrophy as the cause of
the correlated increase of the
other parts. In fiddler-crabs,
for instance, and other Crus-
tacea, it might be suggested
that the increased weight of
the large chela caused extra
strain to be put on neigh-
bouring limbs, which then re-
sponded by functional hyper-
trophy. It would be difficult
to reconcile this with the
correlated decrease of the limbs
anterior to the highly heter-
ogenic appendage, which will
be discussed later in this chapter, but the interpretation is
wholly ruled out in a holometabolous insect, in which the
male legs have never had to support the weight of the
enlarged mandibles before they appear in their definitive,
1 Champy's figure (1929, p. 198) of the beetle Oryctes rhadama
indicates a similar gradient with high point anteriorly, but here affect-
ing {inter alia) two sexual heterogonic organs, the two ' horns ' of the
male. The heterogony of the more anterior, cephalic horn is clearly
greater than that of the thoracic horn.
2 Some of the quantitative data for the thorax do not fit in with the
idea of a uniform gradient. It will be necessary to make additional
measurements on other species to clear up this point : as previously
indicated, however, there is no necessity to suppose that such gradients
are always uniform and not complex in shape.
k
2 Z
\°
■
2 0
■
18
■
1-6
•
1-4
:
1?
Head \
1 Thorax \ Abdomen
10
08
?,'"
"XT""--- ;
/
\^~ ^
Fig. 63. — Probable growth-gradients
in the body of male and female stag-
beetles (Liicamts cervus).
n6
PROBLEMS OF RELATIVE GROWTH
enlarged form. And further, even if functional hypertrophy
of the legs were possible in a beetle, it is impossible to see how
it could come into play in regard to the antennae.
The second point is this — that the gradient is concerned
with spatial position, not with morphological position. The
mandibles are spatially the most anterior portion of the body.
Although the antennae are morphologically anterior to them,
they are spatially posterior, and are affected not by a correlated
decrease, as appears to be the rule for organs anterior to a
growth-centre (p. 122), but by a correlated increase.
Another case in which analysis of existing measurements
has permitted us to plot the growth-gradients along the body-
axis, giving us what may
D e^f be called the growth-
profile of the animal as a
whole, is that of the
metamorphosing herring,
Clupea harengus (Hux-
ley, 193 ib ; data of Ford,
partly unpublished,
partly in E. Ford, 1930).
In this case the growth-
gradients are distinctly
complex. Their nature
will be seen from Fig. 64.
Through their operation,
the elongated larval her-
ring is converted, while
considerably increasing
in absolute size, to its
post-larval form. Doubtless if measurements had been made
vertebra by vertebra, instead of at a few points only along
the body, the graded effect would have been more obvious,
and interesting results would have emerged as to what happens
at the boundaries of regions marked by different growth-
coefficients. We may perhaps assume that each region would
possess its own growth-gradient. This seems clearly so in
regard to the head.
Later work by Ford (1931A) indicates that changes similar
in principle but different in quantitative detail are at work
in the metamorphosis of two other species of the same genus,
the pilchard {Clupea pilchardus) and the sprat (C. sprattus) ;
but growth-profiles have not been constructed for these. In
Fig. 64.-
-Growth-profile of metamorphos-
ing herring.
Above, outline of larval herring at onset of metamor-
phosis, showing regions measured (after Ford, 1930) ;
below, its growth-profile, dorsal and ventral : the values
are growth-coefficients (k), relative to body-length.
QUANTITATIVE ANALYSIS 117
addition the same author (Ford 1931B) gives data on the
metamorphosis of the common eel (Anguilla vulgaris). Here
the tail shows positive heterogony during the process. An
analysis of Ford's table (p. 989) shows that when tail length
is plotted against rest-of-body length, its growth-coefficient
is at first over 1-4, then falls at first rather rapidly, then more
slowly to about i-i. It is, however, probable that it exhibits
a growth-gradient with centre of maximum growth anteriorly,
so that to get accurate figures for growth-coefficients we should
need data on a number of separate regions.
It is clear that regions of different growth-coefficients will
often, though not necessarily always, be characterized by
special morphological differentiations. Each such region would
be characterized both by the possession of its own growth-
gradient and its own morphogenetic field (Guyenot). When
this is so, we may expect the region to be capable of extension
over more or fewer body-segments, and to cover a different
serial region of the body-segments in different individuals or
species. This would appear to be the case in regard to different
regions of the vertebral column (e.g. thoracic, sacral, cf. Bateson
1894). A good example of such a shift during individual
development is provided by the Copepods (cf. Caiman, 1909,
p. 89), in which a broad cephalothoracic region is sharply
marked off from a narrow abdominal region. The delimitation
of these two regions is not at a constant position in the body,
but moves back one segment at each moult, either for two
or for three successive moults. It would be interesting to
study the growth-relations of the segments which are thus
transferred from one region to the other : we may hazard
that they would be found to show a sudden alteration of
growth-intensity after their morphological changes, so as to
fit in with the growth-gradient of the anterior instead of the
posterior region.
An interesting example of a general growth-gradient is
described by Faure-Fremiet (1930) for Zoothamnium alternans.
In this colonial ciliate there are two kinds of zooids, large and
small, the latter being in the majority. The multiplication
of the small zooids is not uniform ; but to use our author's
own words, ' decreases according to a kind of gradient in pro-
portion with its removal from the main strain.' It is this
growth-gradient which gives the colony its characteristic
form. The details are complex, and must be consulted in
the original paper ; but the existence of a gradient in power
n8 PROBLEMS OF RELATIVE GROWTH
of multiplication among the cells of a Protozoon colony is
noteworthy.
§ 3. The Two Phases of Growth
Before going further we must consider the difficulty men-
tioned on a previous page — that it is hard to understand how
a regular growth-gradient could produce anything but a series
of appendages regularly graded in size, instead of the irregular
alternation of large, small and medium-sized appendages which
is what we actually find. The answer appears to be that it
would inevitably produce such a regular series if it were the
only factor concerned in appendage-growth, but that it is not
the only factor.
I base this assertion on a study of the figures given by
Herrick (191 1) for the development of the American lobster,
by Giesbrecht (191 1) for that of various Stomatopoda, and by
Schmalhausen and Stepanova (1926) for that of the appendages
of the embryo chick.
In all three cases, there appear to be two successive and
quite distinct phases of growth. In the first, the general
form of the part is being laid down, and this process is accom-
panied by very rapid alterations of form, and by marked
histological changes ; in the second, histological changes are
absent or of an entirely secondary nature, and the form-
changes are confined to quantitative alterations in the pro-
portions of the definitive structural plan. And it would appear
that the regular growth-gradients we have been considering
are manifested only (or mainly) in the second of these two
phases, some quite other mechanism being at work in the
first. Thus not only definitive form-plans, but also marked
differences in size, are established in the short first phase,
and effects of growth during the second phase are confined to
a quantitative modification of the already diversified organiza-
tion given at the close of the first phase.1
The second phase is thus in general of less morphological
importance than the first, although occasionally it exhibits
differences of growth-potential great enough to effect very
1 The matter is still further complicated by the so-called law of
antero-posterior differentiation of appendages (and by the exceptions
to it !) ; this, however, merely means that the onset of the second
phase is not synchronous throughout the body. It is also quite pos-
sible that in the first phase, second-phase growth-effects are present,
but are masked by the much more radical first-phase effects (see §§ 6, 7) .
THE TWO PHASES OF GROWTH 119
striking transformations, as in the chelae of Uca, or the abdo-
men of Pinnotheres. None the less, it is of considerable
theoretical importance as revealing the existence of deep-seated
and regular growth-gradients which appear to be in the first
instance correlated with fundamental properties of the animal
body, such as polarity.
Thus the original effects of such gradients upon growth
would be, relative to the growth-process itself, secondary ;
but since all changes of proportions that occupy a considerable
time must depend on its agency, in certain cases its growth-
affecting properties become modified so as to become of direct
importance to the process of growth of parts.
My meaning will be clearer with the aid of an example.
There is (Fig. 61) a progressive decrease in relative growth-ratio
as we pass forward along the head-appendages of Eupagurus,
which obviously results in a slight and continuous change in
their proportionate size, relative to the body and to each other,
as life goes on ; I cannot conceive, however, that these par-
ticular changes in relative size are of any biological significance,
but am forced to regard them as accidental consequences of
the existence of a growth-affecting gradient in the body. The
comparatively enormous increase in relative size of the male
right chela, however (as of the female swimmerets), does
appear to have biological significance ; and here therefore we
must consider that the form of the gradient has been modified
so that its growth-affecting properties become utilized to
alter bodily proportions in an adaptive way.
Although clearly much more work must be done before
the laws, properties and physiological mechanisms of these
two growth-phases are properly understood, general considera-
tions as well as the limited special analysis I have been able
to undertake make me feel that their existence is real enough.
If that is so, we may enshrine the distinction in specific
terminology, and say that, following the period of chemical
predetermination at which the specific fates of different regions
of the embryo (or regenerating part) are invisibly determined,
there occur two further phases, one of tissue differentiation
and the assumption of the definitive general form-plan, and
one of subsequent quantitative growth-changes in proportions
(and presumably also changes due to functional activity).
To the first of these the term chemo-differentiation has already
been applied (Huxley, 1924c) and adopted by others (Gold-
schmidt, 1927 ; Needham, 1931) ; for the next I propose
120 PROBLEMS OF RELATIVE GROWTH
to use the term histo-differentiation, since histological change
would here appear to be the most decisive factor ; and for
the last, the term auxano-diff eventration, since quantitative
growth-changes are now the most significant.1
Histo-differentiation would obviously be at work not only
in the first formation of organ-rudiments, but also during any
radical metamorphosis. This appears to be the case even
when limbs are converted from one structural plan to another
during development : e.g. when in a lobster a limb is con-
verted from a biramous swimming appendage to a jaw, or
even when the pereiopods lose their exopodites. In this latter
case, it is possible that negative heterogony of the auxano-
differentiative type is at work, but acts so quickly that between
one moult and the next the whole exopodite disappears, but
the presumption appears the other way. It would be interest-
ing to study the histology of the different parts of the limb
during the instar prior to the exopodite's disappearance.
§ 4. Growth-changes Correlated with High Local
Growth-intensity
Another indication of the real existence of these fundamental
growth-gradients is afforded by the fact that it appears im-
possible to effect a marked change in the growth-ratio of one
appendage without at the same time effecting slight changes
in the growth-ratios of the neighbouring appendages — in other
words, that a localized growth-change is not in reality fully
localized, but must operate within the framework of the main
growth-gradient of the body, and affect its working. The
proof of this is afforded by the changes in relative size of
neighbouring appendages which are to be found correlated
with marked heterogony of a particular appendage. The
' control ', by the difference from which the magnitude of the
correlated change is deduced, may be provided by contrasting
corresponding organs, either those of the opposite side of the
body in cases of asymmetrical heterogony of a part, or those
of the opposite sex when there is heterogony of an organ in
one sex only. As examples of the former we can take the
male fiddler-crab (Uca) ; and of the latter, those numerous
Crustacea in which one of the pereiopods is enlarged, in the
male only, to form a powerful chela.
1 Most significant, that is, for our present purpose. If we were
more interested in the changes directly brought about by function,
another term would be needed.
CORRELATED GROWTH-CHANGES 121
In Uca, it has been known for some time that other parts
besides the chelae are asymmetrical in males, notably the
carapace and the walking legs, which also are enlarged on the
side of the large chela. The only quantitative data on this
subject appear to be those of Yerkes (1901), which have been
further analysed in this laboratory (Huxley and Callow,
unpublished). Unfortunately Yerkes' measurements apply
only to large specimens (10-15 mm. carapace-length). Over
this range, the large chela is, of course, enormously much
larger than the small (at least twenty times as heavy, from
my data on U. pugnax) : the merus, measured by Yerkes, is
about 60 per cent, greater in length. The only other relevant
measurements taken by Yerkes are (1) the merus-length of
the first walking leg, which is about eight per cent, larger
on the large chela side, and (2) the lateral margin of the cara-
pace, which is rather over 5 per cent, larger on this side.
What most concerns us is that for the merus of the first
walking legs, the excess of the large-chela side increases dis-
tinctly, over the size-range measured, with absolute increase
in size of the animal, from below 7-5 per cent, to over 9 per cent. ;
and that for the carapace margins, in spite of some irregularity,
appears to increase slightly (from below 5 per cent, to 5-5 per
cent.), i.e. the asymmetry is progressive.
Work is in progress by Miss Tazelaar in this laboratory on
this subject. Her preliminary data indicate clearly that all
the walking legs are implicated in the increase of size on the
side of the large chela, that the first after the chela (second
pereiopod) is most affected, the second next, while the effect
on the last two is slight and somewhat irregular ; e.g. six
males of mean carapace-length 14-3 mm. gave the following
result for the percentage excess of the limb of the large side
over that of the small side :
Pereiopod
. A .
Second Third Fourth
Excess on large side, per cent. . 2-6 0-9 0-2
A graded effect is clearly visible.
In Uca, measurements have unfortunately not yet been
taken of the appendages lying just anterior to the chelae.
This has, however, been done in several cases of sexual differ-
ence in heterogony, and in every case we meet with the sur-
prising fact that whereas, as in Uca, marked heterogony of
an organ is associated with a slight increase of relative size
122
PROBLEMS OF RELATIVE GROWTH
in the appendages just posterior to the special growth-centre,
there is a slight decrease of relative size in those immediately
anterior to it.
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o
° Aft-
•2?
60
-
-
50
4 0-
■r
V
A
-
30
20
i
t i r
\\
r ■ i
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Maxilliped
(aid)
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5th.
- 60
-50
- 40
30
20
Chela
Pereiopod
3rd. Chela 2nd.
mxpd propus s*
(xlO) (X6)
B
Zrd.
Pereiopcds
Fig. 65. — Graph of changes in relative size of parts associated with a local
region of high growth-intensity (male chela) in spider-crabs, (A) Maia
squinado, (B) Inachus dorsettensis.
CORRELATED GROWTH-CHANGES 123
This was first discovered in regard to weight-measurements
on the spider-crab Maia squinado (Huxley, 1927) ; and next
confirmed, this time in regard to linear measurements, on
another spider-crab, Inachus dorsettensis (Shaw, 1928). I then
felt that the phenomenon might be due to one of two rather
different causes. In both these cases, the organ diminished
in size in the male was the third maxilliped, the pereiopod
which was enlarged as the male chela being the first of the
walking-leg series. It was thus possible that the effect had
nothing to do with the main growth-gradient, but was in
some way due to the nature of the appendages concerned,
maxillipeds for some reason responding differently from
pereiopods.
To decide between these two alternatives, it was necessary
to find an organism in which some other pereiopod than the
first was enlarged to produce a large chela. The true prawns
provide a case in which the second pereiopod is so enlarged ;
but unfortunately in the common British species there is little
difference between the sexes in the size of the large chelae.
Eventually, through the kindness of Dr. Seymour Sewell, of
the Indian Museum, a number of specimens were obtained of
the magnificent Indian prawn, Palaemon carcinus, a species
with marked sexual dimorphism in the chelae, and measure-
ments on these established the fact that the effect is a true
positional effect, since in the male, whereas the relative size
of third, fourth and fifth pereiopods were increased, that of
the first pereiopod as well as of the third maxilliped were
decreased.
In general, male Crustacea appear to have relatively larger
pereiopods than females. But here the difference between rela-
tive size of pereiopods in the two sexes is much reduced in the
pereiopod anterior to the chela, showing an inhibiting effect
of chela-growth. Another method of studying this pheno-
menon is to take the relative growth-rate of the male and
female appendages. When this is done, it may occur, over
a given size-range, that the relative growth-rates of the organs
anterior to the chela are actually lower in the male than the
female (see Figs. 66, 67).
The explanation of this curious effect is for the moment
completely obscure, though there can be little doubt that it
is connected with the fact that any fundamental growth-
gradient along the body-axis must be polarized.
Effects presumably of the same general nature, though not
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CORRELATED GROWTH-CHANGES
125
upon growth, have been recorded by Gabritchevsky (1930)
in his experiments on regeneration in spiders. In successive
moults, the legs of the species used by him pass through a
r e ad i 1 y -char-
acterized series
of develop-
mental stages.
If a leg be
amputated, its
regeneration is
accompanied by
alterations in
this sequence in
the neighbour-
ing legs. Some-
times the nor-
mal sequence
is accelerated,
sometimes re-
t ar ded ; in
either case,
compensa t o r y
effects in the
other direction
may be subse-
quently pro-
duced. Gabri-
tchevsky's re-
sults to date do
not permit us
to say why
either of these
two opposed
effects may be
found ; we may
anticipate that
the cause will
prove to lie in
the rapidity of
regeneration and the time before moulting at which the oper-
ation was carried out.
Przibram (1917) carried out an elaborate investigation on
regeneration of legs in the Mantid Sphodromantis bioculata.
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pereiopods.
Fig. 67. — Correlated changes of growth-rate in struc-
tures immediately anterior and posterior to a local
region of high growth-intensity (male chela) in the
prawn, Palaemon cardials.
The graph represents the growth-rate (% increase for 100% increase
in carapace length) of various appendages in males and females.
126 PROBLEMS OF RELATIVE GROWTH
His results can be analysed (Huxley, unpublished) to shed
some light on our problem, although not so much as if the
experiments had been designed for that purpose.
His results are particularly valuable in one respect, as the
growth-rate of the regenerating and normal limbs was followed
during a number of moult-stages, until the animals became
adult (or died). In general it appears that after amputation
of the fore or middle leg, the normal growth of the next pos-
terior legs is first depressed during the period of most active
regeneration ; then accelerated ; and finally, as the rate of
regeneration approximates to that of normal growth, sinks
again, usually below normal. (The rate of growth of the
regenerating limb also frequently sinks below normal at about
the same time.) The effect on the unoperated leg of the
opposite side of the same segment as the amputated leg appears
to be similar, but is less pronounced. After amputation of
the hind-leg, the effect on the next anterior (middle) legs is
of the same general form, but the initial depression is greater,
the later acceleration less. (I have no data for the effect of
amputation of the middle leg on the growth of the fore-legs.)
When, as sometimes happens, no regeneration occurs, the effect
of amputation appears to be a temporary acceleration of
normal growth in other legs, without any marked depression
at all. The effect on limbs in other segments is always identical
on both sides ; the effect is no greater on the side where regen-
eration is in progress (Fig. 68).
Thus the effects depend partly upon the rate of growth in
the regenerating limb ; but there appears to be also a positional
effect, the depressant effect of a regenerating (rapidly-growing)
limb being more marked on the limb anterior to it, the stimu-
lative effect more marked on the limb posterior to it. This
fits in with the facts of normal growth in Palaemon, Maia and
Inachus.
Von Ubisch (1915) carried out somewhat similar experi-
ments on the larva of the insect Cloe diptera. Unfortunately
his method of recording his results does not permit of a satis-
factory analysis of the question which here interests us, namely
the effect of regeneration on the normal growth of neighbour-
ing unoperated structures, but only of effects on the rate of
regeneration itself. However, some of his results are of
relevance. All operations were made directly after one moult,
and the final measurements made directly after the next
moult. He finds that when two different legs are removed,
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128 PROBLEMS OF RELATIVE GROWTH
their rate of regeneration is greater when both are on one
side of the body than when they are on opposite sides ; and
the same holds good when three different legs are removed. The
mean difference amounts to nearly 5 per cent, of the lower figure.
Further, if two legs are removed, regeneration of a given
leg is greater (by nearly 3-5 per cent.) if the other leg is the
opposite member of the same pair than if it is a leg of another
pair, whether on the same or the opposite side. On the other
hand, when only one leg of a pair was amputated, its normal
partner on the opposite side was not accelerated in growth,
but if anything slightly retarded (mean, 0-4 per cent, retarda-
tion— a figure which may however well fall within the limits
of experimental error). This result cannot be compared
directly with the corresponding results in Przibram's Sphodro-
mantis, since it only concerns growth during one instar.
Going through his tables, however, I have come across one
fact which appears to be significant for our problem. When
legs belonging either to two or three different segments were
amputated, whether on the same or opposite sides, the amount
of their regeneration can be compared with the amount made
by the same legs when only a single leg is amputated. It is
then found that the anterior of the regenerating legs always
regenerates relatively less rapidly than the posterior (only one
exceptional result in eight series of experiments) ; in five out
of the seven concordant series, the regeneration of the anterior
limb is actually below the ' normal ' amount for regeneration
of the corresponding single limb, and that of the posterior
limb equal to or greater than normal. This again points to
a depressant effect of rapid growth on anterior structures,
and a stimulative effect on posterior ones.
It will, however, be necessary to carry out systematic experi-
ments, using repeated regeneration, before clear-cut results
can be obtained.
§ 5. Some Cases of Teratological and Abnormal Growth
In this connexion, the measurements made by Nanagas
(1925) on human anencephalic foetuses are of interest. In such
monsters, the size and proportions of the trunk and especially
of the lower limbs, are practically normal, but, while the cranial
region is, of course, markedly underdeveloped, the fore-limbs
are hypertrophied (by about 12 per cent.). Furthermore, the
hypertrophy is graded (Nanagas' Table I) : the upper arm
shows an increase over the normal of 24 per cent., the fore-
ABNORMAL GROWTH 129
arm of 16 per cent., the hand of 7-1. 1 The length of the middle
ringer is actually decreased relative to normal. There is thus
not only a growth-gradient in the arm, but an indication that,
as in male stag-beetles (p. 115), the excessive tilting of one
end of the gradient above normal has led to a slight depression
of the other end below normal. The fact that the proportions
of the segments of the lower limb are not affected is interpreted
by Nanagas as meaning that the disturbance to growth occurred
very early, before the differentiation of the hind-limb had
begun. There appears, to be, further, a slight graded effect
upon the trunk. The arm circumference is 13-0 per cent.,
the chest circumference at the nipples 7-5 per cent., at the
tenth rib 1-3 per cent, above normal. In any case, the corre-
lation of poor development of head with increased development
of the region next behind it is interesting. It may be inter-
preted as due primarily to the excess growth of the forelimb
region ; or more probably the primary factor is the failure
of the head to develop, which then, in accordance with the
views of Child and of Stockard, permits a higher development
of regions posterior to it in the body-gradient (Fig. 69, left).
A further teratological study which is also relevant to our
purpose is that of Mead (1930), based on a single monster of
34-8 cm. crown-heel length. When the measurements were
compared with those computed for a normal embryo of the
same crown-heel length, it was found that the head was very
much enlarged ; likewise the arms, in diameter more than in
length. The trunk was enlarged in breadth, more so anteriorly
than posteriorly ; its anterior portion was markedly reduced
in length, its posterior portion slightly increased. And the
hind-limbs were slightly increased in length, but considerably
reduced in diameter. (The proportions of the segments of
the limbs were not disturbed.) This can most simply be inter-
preted as due to (1) an abnormally high growth-centre anteri-
orly with growth-gradient extending backwards. But (2) this
is complicated by differences in growth-intensity in the different
planes of space. From the shoulder region posteriorly there
is an abnormal growth-gradient in the breadth-dimension,
above normal anteriorly, below normal posteriorly (cf . the arm
of Nanagas' specimen, supra) ; and growth in the length-
dimension is inversely correlated with growth in the breadth-
1 This is the statement in Table I, and appears to be correct. On
p. 477, however, the value is given as i-i per cent., and on p. 485
as 2 per cent. !
9
130
PROBLEMS OF RELATIVE GROWTH
dimension, so that the chest is much shortened, the legs
slightly lengthened.1 For some reason, this inverse correla-
tion between breadth- and length-growth does not seem to
have affected the head region (Fig. 69, right).
While many details remain obscure, these cases at least
provide excellent examples of graded abnormalities of growth-
intensity. Professor C. R. Stockard has drawn my attention
to a similar case, but affecting an isolated organ only. Streeter
Fig. 69. — Graded-growth effects in two human monsters.
(See text for details.)
(1930), in his Plate 12, figures a case of isolated hypertrophy
of a single human finger. The point which interests us, how-
ever, is that the proportions of the phalanges appear to be
abnormal, and to have been affected in a graded way.
While on the subject of abnormal growth, we may refer to
some cases of dwarfing and gigantism. Sir Arthur Keith has
kindly let me see some MS. notes on this subject. There are
1 This interpretation is not identical with that of Mead, who pre-
fers to think mainly in terms of the ' law of developmental direction '.
ABNORMAL GROWTH 131
also numerous cases scattered in the Nouvelle Iconogmphie de
la Salpetriere, and Stockard (1931) treats of the question in
a comprehensive way. In general, it appears that simple
pituitary gigantism is associated with relatively long limbs,
but within the limbs there is not much proportionate change,
or a slight defect in hand and foot. In acromegaly, however,
whether associated or not with gigantism, the relative sizes
of hand and foot are increased.
In ateliotic dwarfs, the proportions of the limbs to the
trunk and of the limb-segments to each other are not affected.
In achondroplastic dwarfs, however, not only are the limbs
abnormally short, but the segments are differentially affected ;
the proximal segments are markedly reduced, the hand and
foot scarcely at all. (In mongoloid dwarfs, on the other hand,
relatively short fingers and toes are among the most striking
differential characters (Davenport and Swingle, 1927).) The
hind-limb is usually less abnormal than the fore-limb. In
breeds of dogs with markedly reduced limbs, which Stockard
(1. c, p. 228, etc.) compares with human achondroplasia, ascrib-
ing both to abnormalities in the thyroid, we get a similar
modification of the proportionate size of the limb-segments.
In the basset-hound he states that the hind-feet are dispro-
portionately enlarged, which would indicate a tilting of the
whole gradient so that the proximal region is below, the distal
region above the normal level (cf. pp. 89, 115). Unfortun-
ately he gives no precise details.
The relatively large size of limbs in giants can only be due
to their increased heterogony during growth ; while achondro-
plastic types afford evidence of a disturbance in the growth-
gradient as well as the general growth-coefficient of the limbs.
From Stockard's work it is clear that, in dogs, the achondro-
plastic short type of limb may be inherited separately; it
depends upon a single Mendelian gene.1
1 The apparently genetic achondroplasia (chondrodystrophy) found
in fowls (Landauer, 1927) differs in its results from that of mammals
in various ways. In the first place it does not affect the fore-limb,
while the hind-limb and most other parts of the body are markedly
affected. Secondly, in the hind-limb the tibia is more affected than
the femur. In the pelvis, growth in length is little affected, but many
breadth-measurements are enlarged. In a dwarf six-months-old
chicken whose dwarfism was apparently of myxoedematous origin,
Landar (1929) comes to the conclusion that its deviations from normal
proportions (which were negligible in the wing, but marked in the leg,
skull and pelvis) can mainly be attributed to a suppression of the
later phases of growth.
132 PROBLEMS OF RELATIVE GROWTH
§ 6. The Law of Antero-posterior Development and
its Effect upon Growth
In this connexion, we have the important work of Scammon
(references in Scammon and Calkins, 1929) on the growth of
the human foetus, from about 5 cm. length to birth. He there
finds definite evidence of gradients as regards growth, proceed-
ing antero-posteriorly along the main axis, and centripetally
along the limb axes. This main gradient is found for internal
organ-systems (gut, vertebral column, etc.) as well as for
external form. There are a few exceptions (e.g. the sacral
region of the vertebral column) which apparently, like in-
tercalated centres of high growth-ratio in Crustacea, such
as the large chela of male Uca, are concerned with special
adaptive growth of particular organs.
However, he asserts that practically all parts so far measured
grow in linear proportion to crown-heel length, according to
the simple formula y = ax + b. He further points out that
when b is zero, the percentage measurements of the organ,
relative to standard length, remain constant throughout the
period. If, however, b is positive, the percentage measure-
ments decrease with increase of absolute size, while the converse
is the case if b is negative. Since the anterior regions have b
positive, while in the posterior regions it is negative, there
is the appearance of a growth-gradient. However, this would
not be brought about, as in my previous examples, by differences
in growth-intensity of the various parts, as measured by
constant differential growth-coefficients, but according to
whether the organ in question had made much or little growth
during the embryonic period (below 5 cm. length). This he
would interpret as due to the Law of Developmental Direction,
according to which anterior (and proximal) regions are formed
first, are soonest through with their origin and histological
differentiation, and can embark earlier on their main growth-
period. As we may presume that the growth of each separate
organ follows the usual rule for the body as a whole, namely
that the (compound interest) growth-rate slows off progres-
sively from the beginning of growth, we should expect that
there would be a lag between anterior and posterior regions,
such as that at any given moment the anterior ones would be
at a later and therefore slower phase of their growth than
those lying more posteriorly (and see §7).
It remains to be seen whether this will account for the whole
ANTERO-POSTERIOR DEVELOPMENT 133
of the difference. Unfortunately, Scammon's final extensive
data have only recently come into my hands, and time has
not yet been available for their full consideration and further
analysis in relation to the ideas set forth in this book.
I incline to the view, after preliminary inspection, that
some of Scammon's data would be much better fitted by an
expression of the heterogonic type, y = axk, than by the linear
formula he adopts. (It must, of course, be remembered that
his linear formula, when b = o, is a special case of the heter-
ogonic formula.) In any case, his analysis is valuable in show-
ing that the time-relations of development during the early
embryonic period of histo-differentiation, can exert a marked
effect on the percentage changes of parts in the later foetal
stage when auxano-differentiation is in progress.
Latimer and Aikman (1931), in a study of the prenatal
growth of the cat from total weight 0-3 g. (264 specimens,
including 35 newborn), give further interesting data.
The formula for the growth of the weights of various organs
(y) against total weight (x) are found to be as follows (all in
grams) :
Head . y = x° '97 — 0-69^
Trunk . y = 0-59^ — 0-36 from 1 to 70 g. total wt.
y = x108 — 0-84* + 0-9 from 70 g. on.
Fore-limbs (2) . y — o-o8# from 1 to 90 g.
y = 0-053^ -j- 2-43 from 90 g. on.
Hind-limbs . y = o-i*11 — 0-053 from 1 to 100 g.
y = 0-07,1; -f 3-5 from 100 g. on.
Unfortunately in their tables the authors only give values
calculated on these empirical formulae, and the percentage
curves for relative weights in their figures do not appear
always to be consonant with the formulae. The percentage
curves are reproduced herewith. The constant decrease in
relative weight of the head, constant increase of that of the
trunk, and increase to a maximum followed by a decrease
for the limbs, is clearly brought out (Fig. 70).
For an analogous case on invertebrates of the marked
effect which the law of antero-posterior development can exert
upon bodily proportions, reference may be made to the measure-
ments of Seymour Sewell (1929) on Copepods. During the
later free-living copepodid stages of development, before the
adult phase is reached, the number of abdominal segments is
increasing owing to the division of the less-differentiated sub-
terminal region of the abdomen. Growth is proceeding at
134
PROBLEMS OF RELATIVE GROWTH
160 180 EOOgm
X
75
70
60
50
40
30
120 140 IbO ISO 200jm
100 120 140 160 IBO ZOOjm
D
Fig. 70. — Changes in relative weight of vari-
ous organs of the cat during pre-natal life.
(A) Head ; (B) trunk ; (C) fore-limbs ; (D) hind-
limbs ; all against total weight.
the same time, and in
such a way that, as Sey-
mour Sewell, says (1. c.,
p. 9), ' when a segment
of the body divides into
two, as for example in
the development of the
abdomen, the total pro-
portional length of the
two daughter-segments
is always greater than
the proportional length
of the parent-segment '.
As result, the propor-
tional length of the ab-
domen increases steadily.
But this change in rela-
tive size is quite distinct
in character from the
change in proportional
size in different regions
of the antennae of the
same animals (p. 85),
in which the definitive
number of segments has
been differentiated before
the growth-changes oc-
cur. Many ontogenies
would undoubtedly yield
interesting results if ana-
lysed quantitatively in
the light of the principle
of heterogony on the one
hand and of that of
antero-posterior develop-
ment on the other. This
would apply especially
to ontogenies of primi-
tive type showing con-
tinuous gradual change,
such as those of Trilo-
bites (cf. Raw, 1927).
It is possible that the
ANTERO-POSTERIOR DEVELOPMENT 135
facts obtained by Przibram (1917) for the growth of limbs
in Sphodromantis are also a consequence of the law of antero-
posterior development. His Table B gives the growth-quotients
for the middle and hind limbs at each instar — i.e. the ratio
of the length at one instar to the length at the instar pre-
ceding. If these are averaged for groups of three moult-stages,
we obtain the following result :
Growth-quotients
Moults Mid-leg Hind-leg
2-5 . . . . . .1-211 I-2o6
5-8 ...... I-26o 1-266
8-11 ...... 1-293 1-297
i.e. not only is there a steady increase in the growth-quotient
during life, but the growth-quotient for the mid-leg begins
higher but ends lower than that for the hind-leg. There are,
however, considerable irregularities in the values for single
instars, and it is possible that this result is not significant.
That the law of antero-posterior development depends upon
some fundamental gradient within the body as a whole is
indicated by the work of Ruud (1929) who found that the
growth of a urodele leg-bud rudiment transplanted to the
arm-region was markedly accelerated, that of an arm -bud
rudiment transplanted to the leg region retarded.
Regeneration as well as growth may be affected by this law.
For instance, Von Ubisch (1923) finds that if three equal
V-shaped pieces are cut out of the dorsal fin of Urodele larvae,
the regeneration of the anterior piece is almost always less
than that of the posterior. This latter result he ascribes to
the capacity for regeneration being inversely proportional to
tissue-differentiation, and to differentiation proceeding in an
antero-posterior direction.
In Cloe, the capacity for regeneration remains unimpaired
throughout life, independently of differentiation which is com-
plete in all three limbs, so that we presumably obtain a direct
effect of the main axial gradient of the body upon rate of
regeneration.
However, that the law of antero-posterior development, as
regards growth-effects, need not hold even in mammals is
shown by the measurements of whales given by Mackintosh
and Wheeler (1929 : see especially pp. 277-95). In these
animals the head is, of course, enlarged to carry out the highly
specialized straining function of the baleen on the jaws : and
accordingly we find that its percentage length relative to total
136
PROBLEMS OF RELATIVE GROWTH
length increases steadily from small foetuses to large whales
25 metres long (at extreme sizes it appears to fall again slightly).
Similarly, the relative tail-length decreases with absolute size,
from the juvenile stage onwards. The results are essentially
similar in Blue and Fin whales (Balaenoptera musculus and
B. physalis). Measurements of two dimensions are herewith
given for male Blue whales.
TABLE XII
Relative Head- and Tail-lengths in Male Blue Whales (Balaen-
optera musculus) from S. Georgia (from Mackintosh and Wheeler,
1929
Relative
Length,
No. of
Relative
head-length
Absolute
head-length
No. of
tail-length
per cent.
metres
specimens
per cent.
(tip of snout
to blowhole)
(calculated)
metres
specimens
(notch of
flukes to end
of ventral
grooves)
(r 2
5
J4-45
0-215
5
45-67
Foetus a
7
1476
0-369
8
44-06
|3- 4
3
1536
o-537
3
43-28
U- 5
1
15-16
0-683
1
4379
17-18
13
15-81
277
9
44-95
18-19
14
16-04
2-97
12
44^5
19-20
18
16-58
3-24
13
45-09
20-21
14
17-16
3"52
8
43'8i
21-22
16
17-49
376
15
43-32
22-23
21
17-72
3'99
18
43-43
23-24
38
18-44
433
26
42-91
24-25
56
19-06
4-67
39
41-24
25-26
25
18-72
4-78
19
41-85
26-27
3
19-01
5-04
3
41-98
This is precisely the opposite of what is found by Scammon.
Clearly the head is endowed with a specific heterogony which
counteracts the effect of its precocious development, and, from
the foetal stage on, overrides the effect of the tail's later
differentiation.
In passing, it may be mentioned that the measurements
given by Mackintosh and Wheeler could furnish a rich mine
of information for constructing growth-profiles for both sexes
of the two species of whale. Unfortunately, only percentage
measurements are given for the means, and without mean
absolute measurements for the total length classes. For
accurate work it would accordingly be necessary to recalculate
the original absolute data in the appendices.
ANTERO-POSTERIOR DEVELOPMENT
137
total length, m.
17 20 23
To obtain accuracy I have recalculated the absolute data
for head-length on the assumption that the mean total length
for Class 1-2 m. is 1-5 m., and so on ; the figures appear in
Column 4 of the table. A log-log graph from these (for post-
natal life) is given in Fig. 71. Up to 25 m. total length, it
shows a remarkable approximation to a straight line, giving
a growth-partition coefficient of head-length relative to total
length of about 1-55. When the figures for foetal life are
plotted, they also show a
straight line, but indicating
a growth-coefficient of only
about 1-05. The two lines
intersect at about 17 m.
total length ; if this gives a
correct indication, it means
that the marked heterogony
of the head does not begin
until long after birth, which
appears to occur at 6-7 m.
length, but before sexual "«
maturity, which comes on
at 22-23 m- length.
Numerous further excep-
tions can be found among
the limbs of Arthropods.
As a simple example, we
find that in primitive Cope-
pods both the first appear-
ance and the subsequent
growth of the appendages
occur in strict antero-posterior sequence ; but in many of the
more specialized members of the group, while the appear-
ance of the rudiments still takes place in this sequence,
the swimming feet of the anterior thoracic segments then
grow rapidly, while the more anterior maxillae and maxil-
lipeds remain for some time in a rudimentary bud-like con-
dition (see Caiman, 1909, p. 89). Scammon himself mentions
some exceptions to the purely linear growth of parts. The work
of Schultz (1926, 1930) 1 on other Primates refers to various
1 Schultz presents his results almost entirely in the form of percentage
values (ratios) and their changes with age. It would be of considerable
interest to undertake an analysis of his absolute data to see whether
they conformed toScammon's linear or to my heterogonic formula.
Fig. 71 . — Head length against total length
in male Blue-whales ; logarithmic plot-
ting, k = 1-55 (except for last point).
(From Data of Mackintosh and Wheeler, 1929.)
138 PROBLEMS OF RELATIVE GROWTH
evolutionary changes in proportion of parts which are difficult
to account for solely on variations in the time-relations of early
development. (E.g. increase in relative length of arm without
a corresponding change for the leg : the fact that the ratio
of radius/humerus increases during ontogeny, and increases
most in those animals in which it is highest in adult life, which
are, further, those with the greatest relative length of the
arm). Further, the work of Hammond on sheep previously
cited (p. 88) is conclusive proof that growth-gradients as defined
by me do exist in vertebrates and may even change their
sign, late in development, long after the embryonic period
is over ; while the disturbances of proportions which occur
in conditions due to glandular abnormality (e.g. acromegaly)
cannot originate in the embryonic period when there are no
functional ductless glands, but must be due to changes in
growth-intensity of parts during foetal and post-natal life.
And the analysis of Lapicque (1907) and Dubois (1922) shows
that the growth of some organs at least, such as the brain,
do not take place according to a linear but to an exponential
function of body-size. My tentative conclusion would there-
fore be that in man as in other forms, the law of developmental
direction is of great importance, exerting effects on propor-
tionate size of parts long after it has actually ceased to be
at work in the early embryonic period, but that heterogony
of parts associated with gradients in actual growth-intensity
also operate during the whole of the later period of auxano-
differentiation. It must suffice here to point out the simi-
larity between these and the results obtained by Abeloos (1. c.) on
Planaria. The existence of an identical type of growth-gradient
in two such remote types as a flatworm and a mammal is striking.
It should also be pointed out that the gradients revealed
by Scammon's figures are capable of modification and even
reversal. In the limbs of man, the gradient in male and
female must clearly be quantitatively different, in order to
produce the relatively smaller hands and feet of the female.
In women, further, the main body-gradient is clearly altered
by the accentuation of growth in the pelvic region, apparently
associated with diminished growth in the region of the shoul-
ders ; and the partition of growth-potential between trunk
and extremities appears to be shifted slightly in favour of
the trunk in the male as in Crustacea, e.g. Eupagurus (Bush, 1. c.)
and Gammarus (Kunkel and Robertson, I.e.). A case of
reversal of the limb-gradient has already been referred to in
EMBRYONIC GROWTH 139
Hammond's sheep (p. 88) : here, before birth, the normal
effect must have been proceeding ; but after birth the distal
regions increase least, the proximal regions most, with a graded
effect in between.
Thus in the vertebrate body again, we would appear to
be dealing with a primary gradient effect, doubtless correlated
with Child's physiological or axial gradients, which auto-
matically has an effect upon growth, and secondary modi-
fications of this, imposed to effect growth in biologically
advantageous ways. (See also Bray's work, p. 259.)
§ 7. The Mathematical Formulation of Relative
Growth in Embryonic Life
Organs developing according to the law of antero-posterior
development are special cases of the more general rule that
during early development different organs do not originate
at the same time. They constitute the most abundant of
such cases, and have particular interest, e.g. in relation to
gradient theories. But from the point of view of formulating
qualitative rules of relative growth, they remain special cases
of the more comprehensive rule.
This problem has only been adequately attacked by Schmal-
hausen (1927A, 1927B, 1930), and in what follows I can do little
save summarize his views and to comment briefly upon them.
It is an obvious fact of observation that organs or parts
of the body in general grow more rapidly when first formed,
and that their absolute growth-rate (when external factors
such as temperature are kept constant) diminishes progress-
ively with time. Our previous method of establishing the
coefficient of relative growth for an organ by comparing its
size with that of some standard representing the measure of
the rest of the body at different absolute sizes, is completely
valid only on the assumption that the organ and the standard
part begin their careers simultaneously, so that the decrease
of absolute growth-intensity proceeds pari passu in both,
and time can therefore be neglected. Even when the origins
are not simultaneous, it will, however, usually provide
a close approximation when we are dealing with the later
stages of growth, for then the difference in time of origin
between organ and standard will be negligible in comparison
with the time that has since elapsed. This point is brought
out by Schmalhausen himself, who gives some theoretical
calculations on the subject (1927B, p. 41, etc.).
140 PROBLEMS OF RELATIVE GROWTH
But during the early stages of development, the effect of
different time of origin will be relatively large, and will com-
pletely vitiate the method of comparing absolute sizes (I.e.,
p. 59). What we require to do, if the organ x is first formed
n days after the standard part y, is to compare the size of the
organ at n, n + I, n + 2 . . . days with the size of the
standard at o, 1, 2 . . . days : from these sizes, the growth-
coefficient of the organ could be correctly calculated according
to our heterogony formula. But to arrive at these values, we
require to know the time-relations of early development, which
is precisely what we have been able to neglect, with such
economy of time and labour, in our previous approach.
Unfortunately, owing firstly to the difficulty of estab-
lishing the true time of origin of development, and secondly
to variations of developmental rate among individuals, which
make different embryos arrive at the same developmental
stage at different absolute times, accurate time-relations are
hard to establish for embryonic life, and not always service-
able even when established. These difficulties are extreme in
the chick, but serious even in mammals, where, e.g., litter-
size has a marked effect on foetal size and consequently upon
foetal differentiation. For these reasons, Schmalhausen uses
an indirect method for estimating true developmental age.
In his previous papers, Schmalhausen was able to show
that in the chick and apparently in various other vertebrates,
the linear growth of the embryo, measured by the cube root
of its weight, typ, remained approximately constant through-
out embryonic life. Since the specific gravity of the embryo
is very close to 1, then if the weight p is taken in milligrams,
typ can be taken as giving a value in millimetres ; and the
constant rate of growth can be expressed in mm. per day.
Thus the cube root of the weight of the embryo can be taken
as giving a measure of its age.
This approximate constancy of linear growth-rate holds also,
according to Schmalhausen, for the separate organs of the
body. In all cases, there are considerable oscillations in the
value of linear growth per day ; and sometimes the value alters
progressively during development, so that the method can only
be considered an approximate one. None the less, for studies
of relative growth, the method appears to be at least as suit-
able, and certainly much less difficult to arrive at, than
accurate time-measurements.
Starting from these assumptions, we arrive at the following
EMBRYONIC GROWTH 141
line of argument. We wish to find the growth-quotient q of
an organ — i.e. the ratio of the growth-rate of the organ during
a particular phase of its development to that of the body,
or of a standard part representing the body, not during the
same period of time, but during the corresponding phase of
its development.
Schmalhausen had previously established the following
formula for finding what he calls the true growth-rate (C„) of
an organ for a period from time t to time tu during which
the weight (or volume) of the organ has increased from v
to vx. His formula is
_log^x -log v ..
U~ 0-4343 Ci-*) W
Correspondingly the true growth-rate of the body during
the same period will be
log^-log* (la)
0-4343 (h-t) K '
If organ and body are in the same phase of development
during this period, then the growth-quotient q is
Q, = logtti -logy . . . . (a)
This is simply another method of writing the heterogony
formula already arrived at by me, and his q is the same as
my k.
For purposes of dealing with embryonic organs, we should
take t as t0, the time at which the organ and the body begin
their growth, which in this case we have assumed to be at the
same moment. We should then write v and w, v0 and w0
respectively. Formula (2) can then be written :
Log vt= q log w1 + (log v0 — q log w0) • . (3)
Now log Vo — q log Wo will always be the same ; let us call
this expression b.
Then vx = bw^.
As stated above, q is here identical with k in my formula ;
but we now have a further analysis of my constant b, which
however can only be arrived at if we know the initial size of
the organ and the body-standard.
But if the organ and the body begin their growth at different
times, then the matter is more complex. We want to com-
pare the growth of the organ with that of the body during
corresponding periods of their development. Let tx be the
142
PROBLEMS OF RELATIVE GROWTH
time which elapses between the origin of the body and that
of the organ. Then the length of this period is, by Schmal-
hausen's method, measured by the linear increase of the body :
let this be denoted by Lx. Then if the weight of the organ
at times t and tx be v and vlf the linear dimensions of the
body, not for the same time, but for the corresponding phase
of its development, will be (L — Lj.) and (Lx — Lx) ; and the
corresponding body-weights (since the linear dimension is
derived directly by taking the cube root of the weight) will be
as the cubes of these values.
Thus the growth-quotient q for the organ for this period
of time can be written
a= log Pi -log* _ ( ()
* 3[log (U - U) ~ log (L - Lx)] w
The corresponding formula for linear measurements of the
organ will of course be the same, but with the omission of
the 3 in the denominator.1
As example Schmalhausen takes the length of the parts
of the hind-limb of the developing fowl. Here the develop-
ment begins proximally, so that e.g. the most distal (4th)
phalanx of the 3rd digit begins to develop two days later than
the 1st or most basal.
Taking simply the initial and final values for length between
the 8th (or 9th) and the 21st day, he arrives at the following
result.
Femur
{Phalanx 1
3
4
Origin at linear
size (L) of embryo
mm.
377
5-20
6-87
8-25
8-89
k calculated from
Huxley's hetero-
gony formula
q calculated from
the linear modifi-
cation of formula
(4) above
1-43
i-65
1-75
1-64
1-85
i-li
i-oo
0-84
o-86
1 If we knew the time-relations precisely, the proper formulation of
the growth quotient would be
= log vt - log v
^ " " log (w1 — wx) — log (w — wx)
where wx is the amount of weight added by the body in the period tx,
between the time of its origin and that of the origin of the organ.
Mathematically, it may be pointed out, this is not identical with
expression (4) ; but the latter gives a reasonable approximation.
EMBRYONIC GROWTH
143
It is seen that there is in actual fact (q values) a growth-
gradient in the 3rd digit dropping distally (with possible slight
rise quite terminally again) ; while if we do not take account
of the difference in time of origin, the gradient (k values) is
quite obscured and the terminal digit comes to have the
largest ' growth-coefhcient '.
The change in relative size of various organs of the chick
during embryonic life is shown in Fig. 72, and the actual
growth of some others has been plotted from Schmalhausen's
7 8 3 10 II 12 13 14 15 16 17 18 19 20 21 days
Fig. 72. — Changes in relative size in various organs of the chick during
embryonic life.
B, brain ; H, heart ; F, fore-limb ; M, metanephros.
Note the very different shapes of the curves. This depends (a) on the time of origin of the organ,
(fc) on its relative growth-rate.
data in Fig. 73. The value of Schmalhausen's method is
clearly evident. It means that we cannot discover the true
growth-coefficient of an organ during its early stages without
precise information as to the time-relations of development.
Somewhat unfortunately from our point of view, Schmal-
hausen prefers in general not to work with growth-quotients,
which are our growth-coefficients corrected for difference in
time of origin, but with growth-constants. These are obtained
by multiplying the growth-rate Cv of an organ (see equation 1)
for a given period by the mean age of the organ during that
144
PROBLEMS OF RELATIVE GROWTH
period, since he believes that he has established in his earlier
papers the fact that during development, growth-rate sinks
in simple inverse ratio to time.
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36mm.
4 6 8/0/2/4 16 18 20 22 24 26 28 30 32 34 36 mm
A
Fig. 73. — Weight-growth of various organs in the embryo chick, plotted on
growth-rate, A, against age as determined by increase of v'embryo volume
various organs are indicated in the legend. In
In A, x is weight of lens ; and + (continued to the right and below) weight of fore-limb. In
the portions of development shown the lens increases from 8 x 10-6 g. to 0-0088 mg. ; the fore-limb
from 7-5 X 10-4 g. to 0-54 mg. ; the embryo from 0-87 to 41 g.
{Constructed from the data of Schmalhausen, 1927A, Tables 9, 15, 18 ; 1927B, Table 7.
EMBRYONIC GROWTH
145
We need not enter into a discussion of this point here, which
concerns the problem not of relative but of absolute growth.
In any case, the growth-constants thus arrived at will give us
some real measure of growth-rate ; and further, the relative
growth-constant, which he uses for comparative purposes, is
arrived at directly from the growth-quotient above discussed.
However, for our purpose we may stick chiefly to the growth-
10 II 12 13 14 15 16 17 18 19 20 21 days
arith-log paper so that the slopes of the curves is directly proportional to
(see text) ; B, against actual age of embryo. The scales of weight for the
both, 0 denotes weight of embryo.
In B, + denotes brain-weight (from 001 to 1-02 g.) ; + metanephros weight (from 00014 to
0-13 g.). The lens and the fore-limb originate nearly together, but the latter grows much faster.
The brain starts growth early and then grows more slowly than the whole embryo, the metanephros
starts late and throughout grows much faster than the embryo.
IO
146
PROBLEMS OF RELATIVE GROWTH
quotients, or, as I prefer to call them in accordance with the
terminology of this book, the corrected growth-coefficients.
These corrected growth-coefficients for various organs of
five species of birds are as follows (recalculated from Schmal-
hausen's Table 9) :
Growth-quotients
Relative mass-factor
Brain
Giz-
zard
1-20
o-95
1-27
0-94
1-14
Fore-
limb
Hind-
limb
Brain
1-27
1-26
i-8i
I-I2
1-97
Giz-
zard
Fore-
limb
Hind-
limb
Chick (G alius domes-
ticus) ....
Duck (Anas moschata)
House-sparrow (Passer
domesticus)
Sand-martin (Cotyle
riparia)
Rook (Corvus frug-
ilegus)
0-62
0-67
0-51
o-58
o-55
0-94
o-93
0-94
079
0-84
I-I2
1-19
1-05
o-95
1-02
0-244
0-324
0-369
0-491
o-353
0-316
0-316
o-343
0-456
0-438
0-371
0-294
0-371
0-414
0-386
The heterogony of the brain is always markedly negative,
that of the fore-limb slightly negative ; that of the hind-limb
is always higher than that of the fore-limb ; while that of
the gizzard is the most variable.
The right-hand part of the table concerns another constant
arrived at by Schmalhausen. On the assumption previously
arrived at by him that the growth of an organ can be repre-
sented by the formula
v = (at)k (6)
where v is the weight or volume of an organ, t is the time elapsed
since its origin, and a a constant (if the rate of linear growth
of the body or organ is constant, then of course k = 3), then
a is what Schmalhausen calls the extension factor (Extensit-
atsfaktor). The constant a can also be calculated for a given
interval of time (tx — t). A derivative constant is what
Schmalhausen calls the mass factor, m. This he takes as ah.
It can be derived from (6) thus :
,fc —
v
m = cr — -7. , or v
tk'
mtk
(7)
This constant m characterizes the initial size of the organ-
rudiment. Schmalhausen's exposition is here exceedingly
obscure, especially as to how he transforms his absolute
CONCLUSION 147
extension-factor m into what he styles his relative extension-
factor r, which alone makes comparison possible between
different species. It is clear, however, that if his general
argument is correct, the two factors here recorded will serve
to give a complete description of the facts concerning the
relative growth of an organ from its first inception.
We see, for instance, that the brains of chick and duck
start of nearly the same relative size, but that that of the
duck then grows relatively more rapidly. The gizzard is laid
down of larger size in the duck than in the chick, but then
grows much more slowly, so that it ends up considerably
smaller. Similarly, the sand-martin, which in the adult is
characterized by very small hind-limbs, has large hind-limb
rudiments which then proceed to grow very slowly. It is
interesting to find that the growth-quotient of the hind-limbs
is in all species investigated higher than that of the fore-limbs.
I feel that some of the formulae advanced by Schmalhausen
are open to criticism and will need some further corroboration.
However, his method for arriving at the true corrected growth-
coefficient for embryonic organs is of real value, and if possibly
not always fully accurate, is undoubtedly the only way at
present available by which we can arrive at a good first
approximation. (See also the work of Ford, p. 260.)
§ 8. Conclusion
The chief points in this chapter may be briefly summarized
as follows : D'Arcy Thompson's method of employing Car-
tesian co-ordinates to effect the geometrical transformation
of an organism or organ gives evidence of the existence of
orderly growth-changes within the body. These may be of
complex nature, but can be analysed into a series of growth-
gradients. Confirmation of this is provided by quantitative
analysis of various organisms during their growth. A curious
effect is noted by which the presence of a centre of high growth-
intensity intercalated in a main growth-gradient is correlated
with minor changes in the growth-intensity of neighbouring
parts. Those immediately posterior are somewhat increased
in size, those immediately anterior appear to be somewhat
decreased in size : i.e. the main growth-gradient is deformed
in a regular way by the presence of the subsidiary growth-
gradient. Finally, it is pointed out that constant growth-
coefficients of parts and regular growth-gradients within organs
and the body as a whole, such as here described, appear to be
148 PROBLEMS OF RELATIVE GROWTH
operative only during the later phase of growth. During
the earlier phase, when histological differentiation is proceed-
ing, quite other quantitative rules apply. To distinguish
these two phases of development, the term histo-differentiation
is proposed for the former, auxano-differentiation (Greek
avgaveiv, to increase) for the latter.
CHAPTER V
GROWTH-CENTRES AND GROWTH-GRADIENTS
IN ACCRETIONARY GROWTH
§ i. The Accretionary Method of Growth
IN the organs we have so far been considering growth is
essentially of the compound-interest type. That is to say,
the increments of new material produced by growth are
alive, and themselves grow and produce new material in their
turn, so that growth is a multiplicative process. The rate of
growth may, and doubtless does, slow down with increasing
size and age, but this merely means that the multiplying factor
decreases as some regular function of physiological age. The
growth remains of compound-interest type, even if the actual
rate of compound interest is never constant but progressively
decreases.
However, there are many other organs whose method of
formation is radically different, so that their growth is essen-
tially of the simple-interest type. In them, the increments
of new material produced by growth are turned into non-living
material as soon as formed and remain permanently (or until
cast off by ecdysis or other means) in the state in which they
were laid down. They do not contribute any further new
material, so that growth here is not a multiplicative but an
additive process.
Here again the rate of growth may alter with age, but this
only means that the amount of new material added in unit
time steadily decreases ; and the growth remains of simple-
interest type even though the actual rate of simple interest
is continually altering.
We may accordingly distinguish these two types of growth
as the multiplicative, intussusceptive or compound-interest
type on the one hand, the additive, accretionary or simple-
interest type on the other.
The most familiar examples of organs growing by the accre-
tionary method are shells such as those of molluscs, brachiopods,
150 PROBLEMS OF RELATIVE GROWTH
or foraminifera ; but the horns of antelopes, sheep, oxen,
rhinoceroses, etc., as well as the teeth of Vertebrates, also
fall into this category. At first sight the forms engendered
by this type of growth appear so different from those we
have hitherto been considering that we do not even expect
to find that the underlying growth-mechanisms have any-
thing in common. The differences, however, depend almost
entirely upon the basic difference between any multiplicative
and any additive type of growth. It is to my mind one of
the most interesting results of these growth-studies that
we are able to demonstrate the same fundamental fact of
growth-gradients operating to produce these two apparently
unrelated types of organic form.
Let me first take the horn of rhinoceroses as example. It
has been admirably handled by D'Arcy Thompson in his
Growth and Form. I here base myself on his lucid analysis,
which, like so many other important ways of thinking that
enable us to see familiar facts in a new light, seems self-evident
once grasped ; but I add one or two detailed points, and
link it up with the ideas which emerged from the study of
multiplicative growth.
The horn of a rhinoceros, then, is produced by intensive
production of keratin in special form and abundance over a
limited area of the head epidermis. The restriction of horn-
producing potency to a limited area is doubtless of the same
nature as the other restrictions of potency which occur during
early development and sooner or later convert the germ from
a plastic construction capable of marked regeneration to a
determined construction which we can designate as a chemical
mosaic (see Huxley, 1924c). The potency of producing eye,
ear, brain or limb becomes similarly restricted and localized
in the amphibian and other embryo, and the restriction of
horn-potency to a localized area is only another result of this
mosaic-producing chemo-differentiation.
On this horn-area, keratin is being produced so as to
accumulate at right angles to the surface. In addition, the
horn-area itself is enlarging over the surface as the animal
grows. Whether, as seems likely, it is enlarging somewhat
more rapidly than the surface of the head as a whole cannot
be stated with certainty until detailed measurements have
been made ; but this is immaterial to our present purpose.
LOGARITHMIC SPIRALS 151
§ 2. Logarithmic Spirals as the result of Growth-
gradients
If the rate of keratin-production at any one moment were
equal over the whole horn-area, the resultant horn would
clearly have the form of a cone, whose precise shape would
depend upon the relation between the rate of addition of new
material, and the rate of spread of the horn-area over the
surface of the head ; if the two rates were equal, the cone
would be a right-angled one, and so forth.1 But as a matter
of fact, in the common rhinoceros growth is not uniform over
the horn-area : it is at its maximum anteriorly, and grades
steadily down to the posterior margin. As result, the horn
of course curves backwards ; and the precise form of the curve
is that known as a logarithmic spiral.
The properties of this type of curve have been fully dealt
with by numerous authors, and the whole subject ably sum-
marized by D'Arcy Thompson in a series of chapters. I thus
need only remind my readers that the most essential character-
istics of a structure growing in a logarithmic spiral are that
successive increments are all of the same form, though of
increasing bulk (gnomonic growth) ; that the angle which
the tangent to the curve makes with the radius vector of the
curve remains constant ; and that if the spiral grows long
enough to form a number of whorls, the ratio, along a given
radius, of the breadth of each whorl to that of the whorl suc-
ceeding, also remains constant. Further, this logarithmic
spiral form must always result in organisms when (a) growth-
increments are converted into non-living material as soon as
produced ; and (b) there is a constant ratio between the
increments at the two ends of the growing structure, with a
regular (though not necessarily uniform) gradient of growth-
rate between the high and low points. We are thus confronted
once more both with the principle of constant differential
growth-ratios and with that of growth-gradients. Since, how-
ever, these here operate with an additive instead of a multi-
plicative growth-mechanism, the resultant structure remains
of constant (and logarithmic-spiral) form instead of continu-
ously changing its proportions as with a male Uca chela or
a female Carcinus abdomen.
But the rhinoceroses teach us a further highly important
fact. Some species possess two horns instead of only one.
1 The extinct Elasmotherium possessed a horn in the shape of a
flattened cone, with the diameter of its base greater than its height.
152 PROBLEMS OF RELATIVE GROWTH
And in these the second and hinder horn is both smaller than
the first (and also less curved) . This implies that the growth-
gradient made visible in the form of the anterior horn is
continued across to the second horn-area, causing the growth-
intensity to diminish, and therefore resulting in a smaller horn ;
(and also that the shape of the gradient is not constant, but
flattens out, leading to less difference in growth-intensity
between the two ends of the horn-area, and consequently to
a decreased curvature of the second horn). These facts thus
lead to the same important conclusion as did the analysis of
the growth of the appendages of Eupagurus — namely, that
though intensive growth be restricted to specifically limited
areas (there the regions of the limb-buds, here the horn-areas),
yet the agency determining the growth-gradients, whatever
it may be, is organismal, and extends throughout the body.
It can only express itself where the potentialities for intensive
growth exist, but it is itself continuous (as in a rather different,
non-graded way, the hormones are distributed over the entire
system, but only exert effects where they meet with tissues
specifically adjusted to react to them). Thus we must assume
that even in the one-horned rhinoceroses, the growth-gradient
is continuous along the head, but can only reveal itself in
species where a second specific horn-area is present ; and
similarly, that in hermit-crabs the growth-gradient controlling
the relative growth of appendages is continuous, not merely
between the limb-producing areas of successive segments, but
even across large regions in which the capacity for limb-
production has been entirely lost, as in the anterior part of the
male abdomen.
It is worth recalling that we already know of analogous
gradients, and to use a more general term under which gradients
can be included, fields, through the results of experiments on
regeneration and grafting. The mere fact that, normally,
precisely what is lost in amputation is restored in regeneration
points in this direction. The conclusion has been made more
probable by the proof given by Schotte, Guyenot and Weiss,
proof that in regeneration (of the Amphibian limb) the new
tissues are not, as was long held, proliferated from the old, but
are differentiated from a truly indifferent tissue, and will
differentiate normally even if the corresponding tissue has
been removed from the basal stump. And finally, it has been
clinched by the beautiful experiments of Milojevic, Weiss,
Locatelli, Guyenot and Schotte (references in Guyenot and
LOGARITHMIC SPIRALS 153
Ponse, 1930) ; these have shown that if the regeneration-bud
from the tail of a Urodele be removed while still in the
indifferent stage and grafted on to the stump of a freshly-
amputated limb, it will grow not into tail but into limb ;
whereas if it has been left a couple of days longer on the tail
before being transplanted, it would have been irrevocably
determined as tail, and would have become tail even in its
new situation. They can only be interpreted as meaning
that what has conveniently been called a ' morphogenetic
field ' permeates the whole body even of the adult Amphibian.
It normally is without effect — in a sense a by-product, we
may say, of the construction of the animal ; but so soon as
indifferent material is placed under its influence, it reveals
its presence by the effect which it exerts on that material's
differentiation.
Whether growth-gradients and morphogenetic effects on
differentiation are both results of one and the same organismal
field, or whether two essentially different, separate field-
mechanisms are at work, is very difficult to say. Further
discussion of this and related points will be deferred to
Chapter VI.
Thus the horns of rhinoceroses, when considered from the
point of view of relative growth, even without further experi-
mental analysis, reveal interesting and unexpected properties
of the animal body. The same point of view, applied to other
structures of the same nature, is equally illuminating. The
horns of rhinoceroses are median ; we should therefore not
expect to find a difference of growth between their lateral
margins. But as soon as we deal with non-median structures,
we should expect, if growth-fields permeate the animal body,
to find a difference in growth-intensity not only between
anterior and posterior, but also between median and lateral
margins. If this is so, the resultant growth will be in what
is popularly called spiral form, i.e. not merely curving in a true
logarithmic spiral in one plane, but corkscrewing up at right-
angles to the first.
This is what actually occurs in almost all horns of sheep,
goats and antelopes. Sometimes the lateral growth-difference
is very slight, and the ' shear ' at right-angles to the primary
plane of the horns' spiral is scarcely perceptible, as in the
Sable and other antelopes. At other times it is considerable,
and we get the horns of certain sheep corkscrewing out at
right angles to the side of the head. It appears that there
154 PROBLEMS OF RELATIVE GROWTH
is no constancy in the sign of the difference, the excess growth-
intensity being sometimes on the median, sometimes on the
lateral margin. The visible result is that the horns are some-
times coiled clockwise, sometimes counter-clockwise.
Complications, not present in the homogeneous rhinoceros
horn, arise in the Cavicorn ruminants owing to the presence
of the living bony horn-core within the non-living true horn
of keratin. The results of these are analysed in detail by
D'Arcy Thompson, but are not relevant to our present purpose.
§ 3. Growth-gradients and the Shells of Molluscs
The most numerous, various and striking of the structures
which are based on logarithmic-spiral form and are due to
differential accretionary growth are the shells of molluscs.
For the moment, we will omit the special case of the bivalves.
The problems here are identical with those encountered in
the rhinoceros horn, except that the horn is solid and uniform
throughout, the shell hollow. In both cases, form depends
upon constant differential growth-ratios. These are here of
four types : (1) the ratio of growth in length to that in width ;
\J (2) the median growth-ratio ; (3) the lateral growth-ratio ;
(4) the ratio of excess growth at specific arbitrary points to
that manifested in the major growth-gradients.
(1) Constant differential ratio of length-growth and width-
growth. In the absence of any other differential growth
but that between length-growth and width-growth, the shell
(or horn) would assume the form of a cone. The value of this
first ratio determines the form of such a cone. In the rhino-
ceros horn, the physiological mechanisms at work are (a) the
rate of production of horn-substance, (b) the outward spread
, of the horn-producing area. In the hollow mollusc shell,
conditions are quite different, and only one factor is at work,
namely, the angle at which the mantle-edge is inclined to the
main axis of forward growth. This angle is presumably
determined chiefly by the form of the body, and may be
modified by functional differences (see Chapter VI). The
mantle is thus always laying on material in a direction oblique
to the main axis of the shell ; the growth-velocities in length
and in breadth are merely components of this single growth-
function. A high inclination of the direction of mantle-growth
and consequent predominance of the lateral component will
naturally produce flattened cones, to which the shell of the
common limpet is an approximation. A low inclination, on
THE SHELLS OF MOLLUSCS 155
the other hand, will produce elongated cones, as in many of
the early paleozoic Cephalopods.
(2) Constant differential ratio as regards growth in the median
plane. If the length-width growth-ratio remains constant,
but the absolute magnitudes of both components are greatest
at one margin, least at the opposite margin, and are inter-
mediately graded around the two sides of the mantle, our
cone will be distorted, and, so long as the growth-ratios con-
cerned remain constant, will grow into a true logarithmic
spiral in a single plane. Examples are provided by Nautilus,
the great majority of Ammonites, and Dentalium.
Our first growth-ratio will still decide the form of the cone,
but now that the cone is distorted this will be measured by
the ratio of the shell-diameter at any place to the length of
the shell measured from its origin along the curve of the spiral.
The first and second ratio together will decide the tightness
with which the spiral is coiled. There are six main possibilities :
(1) As limiting factor, with median growth-ratio = i-o, or
in other words, no growth-gradient from one end of the median
plane to the other, a cone results.
(2) When the growth-ratio is low, only slightly above unity,
the curve is very slight. In such cases, the mathematical
properties of the logarithmic spiral being what they are, a
many-whorled structure will never be produced, as the radius
of even the second whorl would be of relatively immense
extent, and no organism could do more than produce a portion
of the first whorl. Such forms are realized in the rhinoceros
horn or, among Molluscs, in the shell of Dentalium.
(3) With increasing values of the ratio, the radius of suc-
cessive whorls rapidly decreases. The next possibility is
therefore a shell with more than one whorl, but with no contact
between each whorl and the next. This condition is rare, but
is realized, e.g. in certain Ammonites.
(4) As the ratio increases further, a specific value will even-
tually be reached which allows the outer margin of each
whorl to be precisely in contact with the inner margin of the
whorl following. This condition is not infrequently realized.
The precise value of the median growth-ratio needed to produce
this result will of course vary with the value of the previous
(length-width) growth-ratio. With a narrow elongated cone, a
much higher median growth-ratio will be required just to effect
contact between whorls than with a broader, less elongated
cone (see Fig. 74).
156 PROBLEMS OF RELATIVE GROWTH
As in each case only one specific combination of these two
independent variables will produce such a result, we must
suppose that selection has controlled the precise values to
secure this result, which obviously secures greater strength
than does one in which the whorls do not touch.
(5) The commonest condition, however, which ensures
even greater constructional strength than the one preceding,
is produced by a further increase of antero-posterior growth-
ratio, which has as result the partial overlapping of each old
whorl by the whorls formed later. The degree of overlapping
will obviously vary with the precise value of the ratio (as well
as with other properties of the shell : see below).
A B
Fig. 74. • — Diagram illustrating the co-operation of two growth-ratios in
determining the form of the Molluscan shell. The two plane logarithmic-
spiral shells both have the outer margin of one whorl just touching the inner
margin of the next.
In A the ratio of the distance along a given radius from the centre of the shell to the margin of one
whorl to that of the succeeding whorl is 3-0 ; in B it is 2-0. If there were no lateral growth-ratio
(i.e. if the shells were uncoiled). B would be a more elongate cone than A; correspondingly, the lateral
growth-ratio in B must be higher than in A to cause contact of successive whorls.
(6) Finally, in some cases the growth-rate of the slower-
growing edge of the mantle becomes negligible or even zero,
and the growth-ratio accordingly rises towards infinity. In
such a case each whorl is completely overlapped by all suc-
ceeding whorls. This condition is completely realized in
Nautilus pompilius, and nearly so in Nautilus umbilicatus.
This median growth-gradient may be orientated either way
in respect of the main axis of the molluscan body. In Ammon-
ites, for instance, the high point (growth-centre) seems to have
been ventral, so that the shell curved upwards over the back.
In other cases it is dorsal. In flat-shelled Gastropods (e.g.
Planorbis) the orientation is complicated by the fact of torsion.
THE SHELLS OF MOLLUSCS
157
D'Arcy Thompson (1. c.) has treated the quantitative aspect
of this problem at greater length. Unfortunately, he has not
analysed the whole process, as would biologically be the ideal
method, in terms of two co-operating growth-ratios, but, as
regards part of the problem, he has been content to give a
purely mathematical description. An analysis entirely in
terms of growth-ratios is being undertaken by Professor H.
Levy, of the Imperial College of Science ; meanwhile it will
be useful to give a brief summary of D'Arcy Thompson's
treatment of the matter.
The form of a single curve following a logarithmic spiral
is given by the expression
.,, _ pQ COt a
where r is the radius of the shell from centre to circumfer-
ence ; 6 is the angle of revolution which the spiral has de-
scribed ; and a is the angle between the tangent of the curve
and the radius vector of the curve, which remains constant : this
is known as the constant angle of the curve. In addition, the
ratio of the radii of successive whorls is also always a constant.
The relation between these two constants is given in the
following table (abbreviated from D'Arcy Thompson, p. 534) :
Ratio of breadth
of each whorl to
the next preceding
1-0
1-5
2-0
3-o
5-o
io-o
50-0
ioo-o
10,000
1,000,000
100,000,000
In Nautilus, in all
shells of Gastropods,
usually between 8o°
of successive whorls
decreasing values of
out very rapidly. E
the first whorl were
Constant angle
of the
spiral
. 900
. 86°
18'
• 83°
42'
. 8o°
5'
• 75°
38'
. 69°
53'
• 58°
5'
• 53°
46'
• 34°
19'
• 240
28'
160 52'
ordinary shells, and in all typical spiral
the constant angle is rarely below 8o°,
and 850, and the ratio of the breadth
usually between 3-0 and 175. With
the constant angle, the spiral flattens
.g., if the constant angle were 280, and
1 in. broad, the next whorl would be
158
PROBLEMS OF RELATIVE GROWTH
about 1 1 miles broad : this is what happens in shells like
Dentalium, which represent but a fraction of the first whorl,
and ' never come round ', as D'Arcy Thompson puts it.
Now when we are considering, not a single line describing
a logarithmic spiral, but a conical shell distorted into this
form by growth-forces, the form of the curves described by
the inner and outer margin are identical, but the inner margin
is retarded in its growth by a constant fraction — in other
words, the ratio of their growth-ratios is a constant.1 The
actual retardation can be expressed as the ratio of the length
of the inner margin from the centre of the shell, to that of the
outer margin, at any point. This figure gives the median
growth-ratio we have been discussing (or rather is the reciprocal
of it as we have defined it) . This value can also be calculated
by utilizing the mathematical properties of the logarithmic
spiral (D'Arcy Thompson, pp. 541 seq.). We need not go
into the calculations, but can confine outselves to the results.
The median growth-ratio needed to produce a particular degree
of coiling will vary with the constant angle of the spiral.
We will consider only two cases — the median growth-ratio
needed to make consecutive whorls just touch, and that needed
to produce a shell with spaces between successive whorls, the
breadth of each space being a mean proportional between the
breadths of the whorls which bound it.
Median growth-ratio, of outer to inner border of
shell, needed to produce
Constant angle
« of spiral
(a) Successive whorls
just touching
(b) Successive whorls sep-
arated by a space which is
a mean proportional be-
tween the breadths of the
whorls
890
88°
870
86°
850
8o°
75°
700
65°
I-II
l-24
i-39
i-55
i-73
3-«3
4-27
9-84
18-2
1-05
I-I2
i-i8
1-23
i-3i
i-75
2-32
3-12
4-35
1 Waddington (1929) in various Ammonites finds that this is not
strictly true. The ratio is slightly changing all the time, the formula
for the spirals being of the form r + c = e&e instead of r = e&e.
THE SHELLS OF MOLLUSCS
159
Thus in most Ammonitoid and Gastropod shells with con-
stant angle between 8o° and 85 °, to produce contact between
the whorls the outer border must be growing between 3 and
17 times as fast as the inner. For spirals with lower con-
stant angle (i.e. those with high ratio of breadth-growth to
length-growth), the median growth-ratio must be much higher.
In open-coiled forms, however, the median growth-ratio will
be less, and does not have to increase so fast with decrease
of constant angle to preserve the same spacing. When the
median growth-ratio is very high, we shall get forms whose
whorls completely overlap, like Nautilus, if the constant angle
is high ; but if the constant angle is low we shall then get
types like Haliotis (or most Lamelli-
branch shells).
(3) The lateral growth-ratio. If in
addition to the preceding two differ-
ential growth-ratios, we have also one
in a plane inclined (usually, it appears,
at right-angles) to the median, the result
will be what D'Arcy Thompson some-
what loosely speaks of as a " shear " in
the plane spiral, with as result a ' cork-
screw ' or turbinate spiral — i.e. a spiral
not confined to one plane. This is
prettily shown in the accompanying
sketches kindly given me by Miss M.
Lebour, of the Plymouth Laboratory,
illustrating the origin of this complex
spirality in a larval Pteropod. The
original shell is a hemispherical cap,
produced by growth which is uniform
all round. After a certain stage, how-
ever, a marked difference appears in the growth at the two
ends of the median axis, and a smaller difference in the
growth at the two sides ; and the shell at once begins to
' corkscrew '.
In terms of growth-gradients, what happens appears to be
as follows (Fig. 76) : If ABCD in (a) be the projection of the
growing edge of the mantle, with A the region of maximum
(H — h)> C of minimum ( ) growth, with no lateral
differential growth, then the corresponding gradient is shown in
(c). If, however, a lateral differential is established, the results
will be as shown in (b) and (d). The lateral differential growth-
Fig. 75.
Limacina retro-
versa.
(a) shell of larva i day old ; the
shell is hemispherical ; (b) shell of
larva 3 days old. Differential
growth has begun, (c) Larva in
its shell, 4 days old. Differential
growth has proceeded further, and
the spiral shape of the shell is
apparent.
i6o
PROBLEMS OF RELATIVE GROWTH
ratio is always smaller than the median (if it were larger, it
would of course decide the main spiral, and the other would
become the subsidiary differential, concerned with distortion
of the main spiral).
Whereas the value of the main or median growth-ratio
decides the tightness of the coiling of the main spiral, that of
— c
-- c
©
®
Fig. 76. — Diagram to illustrate the growth-gradients operating to produce
the plane and the turbinate spiral shells of Molluscs.
(a) and (6) Projections of the growing edge of the mantle ; (c) and (d) corresponding elevations,
the ordinates representing growth-intensities, the abscissae distance across the shell-opening, (a) and
(c) gradients operating to produce a plane logarithmic spiral shell. There is a centre of maximum
growth at A, of minimum growth at C. The growth-gradients between A and C are identical on both
sides of the mantle, through B and through D.
(b) and (d) gradients operating to produce a turbinate spiral. The growth-gradient ABC is ofa
different shape from ADC ; thus a secondary growth-ratio is established between B and D. The
growth ratio AP/CS < BQ/DR.
the secondary or lateral growth-ratio decides the degree of
distortion of this spiral, the proportionate amount by which
it is pushed out of its fundamental plane. When the lateral
ratio is unity, the shell is flat, in one plane, like that of Plan-
orbis. When it is low, the shell is low also, like the depressed
THE SHELLS OF MOLLUSCS 161
shells of Helix. When it is higher, the shell becomes more
pointed, as in Turritella. Here again, D'Arcy Thompson
gives detailed mathematical treatment, although for some
reason he has not reduced the degree of ' shear ' (distortion
of the primary spiral in a plane at right-angles to its own)
to terms of differences of growth-rate, as he has done for the
degree of coiling (distortion of the primary cone into a spiral).1
Typically, of course, the shape of any structure produced
by accretionary growth must remain constant so long as the
various growth-ratios concerned in its production remain
constant. But as a matter of fact, the various ratios often
alter with age or size, in some cases suddenly, in other cases
progressively. Thus certain Ammonites have their oldest
portions uncoiled, while the earlier-formed part of the shell
is of typical form — an example of sudden alteration. In
others, the ratio of the diameters of successive whorls does
not remain constant as in the true logarithmic spiral, but in-
creases progressively. This gradual change, due to a progres-
sive change in the shape of a fundamental growth-gradient,
is not infrequent, and may possibly prove to be associated
with senescence. Among Gastropods, the tapering of the
oldest part of the shell in Pupa, Clausilia, etc., in place of the
continued expansion to be expected if the growth-ratios remain
constant, is another case, and there are numerous other
examples.
(4) Finally, there may be excess or defect of growth-ratio
at special points, not in connexion with the main growth-
gradient. This, of course, has its analogy in multiplicative
growth, e.g. in the development of markedly heterogenic
appendages which break the main growth-gradient of the
body, as in the right chelae of male hermit-crabs.
The most obvious examples of such growth are found in
lamellibranch shells. The ' ears ' of the shells of scallops and
other species of Pecten are an excellent case. The general
growth-gradient is of usual Molluscan type, with high point
directly opposite the hinge, and a uniform and symmetrical
double gradient extending thence round both sides of the
shell. Just before reaching the hinge, however, the growth-
ratio, after sinking very low, increases rapidly and then
abruptly descends to zero, thus generating the ' ears ' of the
shell.
1 Interesting numerical details concerning various structures of
logarithmic-spiral construction may be found in Petersen, 1921.
11
162 PROBLEMS OF RELATIVE GROWTH
A rather different example is provided by the razor-shells
(Solen). In these, the normal symmetrical gradient of the
lamellibranch shell, wrdj^grQAvthrcentre opposite the hinge, is
completely distorted by the development of a second growth-
centre at the morphologically posterior (siphonal) margin of
the shell. This occurs, of course, in numerous other forms,
making the shell asymmetrical along the antero-posterior
axis ; but in Solen this ' secondary ' growth-centre has become
more important than the phylogenetically primary one, and
its markedly higher growth-ratio converts the shell into the
well-known elongated blade.1 The margin of the shell at
this end is truncated, but the opposite margin is rounded, the
shell thus consisting of two markedly different but homologous
halves, the one conforming to the normal lamellibranch shape,
the other pulled out to ten or even twelve times the diameter
of the former. Still further examples, again of a somewhat
different type, are seen in the spines which beset the shell of
such forms as the Spiny Cockle {Cardium acidcatum) or various
Gastropods. These represent localized centres of excess
mantle-activity, which, however, are only active periodically.
Permanently^active fluctuations in growth-activity along the
gradient are revealed in such forms which have a crenellated
margin to the shell (e.g. Tridacna). It is not known whether
the presence of such special centres of high growth-ratio is
correlated, as in Crustacea, etc., with slight excess or defect
of growth in adjacent regions.
As in multiplicative growth, these subsidiary regions of
special growth-activity are themselves constructed on the
usual plan, of growth-gradients — single or double — culminating
in a high point or growth-centre.
The bivalve shell (Lamellibranch and Brachiopod) demands
a few words to itself. It_Dwes its form to the existence of
two separate gradients of accretionary growth, each forming
a shell of logarithmic-spiral form. "Typically, the two gradients
are equal but of opposite sign, each symmetrical about a line
drawn from the hinge to the ventral margin ; the growth-
ratio at the hinge is so close to zero as to be negligible, while
the component of growth in shell-width is relatively so large
that the spiral is a very high-angled one, and never forms
even one complete whorl.
1 When distortion of this type occurs, it is usually the posterior
half which is enlarged ; but there are a number of genera which show
the opposite tendency (e.g. Donax).
CONCLUSION 163
This last feature is sometimes carried to an extreme, as in
the lower valve of Pecten shells, where all growth is in the
direction of width, with a perfectly flat shell as resultant.
Or, as in other forms which habitually lie on one valve, the
original direction of the gradient is reversed, and the curva-
ture of the two valves becomes similar in sign, instead of
opposed (e.g. Productus, occasionally Anomia).
Still more complexity of detail is shown in many Pteropod
shells ; these, as well as the shells of Foraminifera, which
achieve logarithmic-spiral form by a somewhat different
method (the addition of whole chambers instead of the mere
prolongation of a single shell), have been well analysed by
D'Arcy Thompson, and need not detain us here. Mention
should also be made of the interesting paper of Sporn (1926),
who applies a different set of mathematical ideas to the analysis
of growth in molluscan shells. These have been related to
growth problems in a more general way by Smirnov and
Zhelochovtsev (1931).
§ 4. Conclusion
We may sum up the most important points of the present
chapter as follows : Accretionary growth, in which the new
material deposited is not itself capable of further growth,
gives rise to structures whose general appearance is radically
different from those produced by ordinary intussusceptive or
multiplicative growth. But the differences turn out to be due
only to this difference in the fate of the new material added
by growth, as result of which the fundamental law of accre-
tionary growth is one of simple interest, that of multiplicative
growth one of compound interest. In other respects, the
relative growth obtaining in the two kinds of structures is
similar. In both we find growth-centres and growth-gradients,
not only major gradients extending through major regions or
the whole body, but also minor gradients superposed upon
and locally overriding the main gradients.
The prevalence of the logarithmic-spiral form in nature is
due to the fact that a uniform single growth-gradient, com-
bined with the method of accretionary growth, must produce
a structure in the form of a logarithmic spiral. Departures
from the strict logarithmic-spiral form are due to irregularities
in the growth-gradients, or to changes, either sudden or pro-
gressive, in one or other of the growth-ratios concerned.
The precise form of the shell or other accretionary structure
164 PROBLEMS OF RELATIVE GROWTH
depends upon the numerical values of the one, two or three
main growth-ratios concerned in the production of conical,
true logarithmic-spiral, or sheared (turbinate) logarithmic-
spiral form respectively.
Thus the essential growth-mechanism underlying the auxo-
differentiation stage of development of a single appendage
(chela, vertebrate limb), a series of appendages (hermit-crab,
stag-beetle), a region of the body (brachyuran abdomen), a
rhinoceros horn, a vertebrate tooth, or a molluscan shell, are
all of the same fundamental nature. In every case we find
constant differential growth-ratios, and these are arranged in
growth-gradients culminating in growth-centres. If we wish
to think analytically about organic form and proportions, we
must think in terms of constant differential growth-ratios organ-
ized in the form of growth-gradients. And if we are ever to
solve the problem of growth physiologically, we should do well
to concentrate on discovering the biochemical basis for growth-
centres and the physiological reasons for the graded distribution
of growth-potential on either side of these centres.
CHAPTER VI
HETEROGONY, GROWTH-GRADIENTS AND
PHYSIOLOGY
§ i. Normal Proportions as result of a
Partition-equilibrium
THIS chapter cannot but be an unsatisfactory one, for
the simple reason that we know so extremely little
about the physiological or biochemical processes
underlying growth in general and growth-gradients in par-
ticular. All that can here be attempted is to bring together
some of the scattered facts and indications which are in any
way connected with the problem.
In the first place, as we have seen in the first chapter, there
are strong grounds for believing that the normal growth-ratio
of a heterogonic organ is in some way determined as result
of a balance between its size and that of the body, equilibrium
being attained when the formula y = bxk is satisfied. When-
ever y < bxk, the growth-ratio of y (the heterogonic organ)
is accelerated (k^. ^>k), and becomes normal once more (= k)
when the organ reaches the size demanded by the original
formula. The organ can become smaller than its size for
growth-equilibrium in a number of ways. The most obvious
is by amputation of the organ, in types where regeneration
is possible. Its regenerative growth-ratio is then much more
rapid than its normal growth-ratio would have been, and
gradually slackens down until it becomes normal with the
attainment of proper relative size by the organ. This is only
a special case, for all organs, whether heterogonic or not,
appear to exhibit the same behaviour during regeneration.
Przibram (1917) has in mantids given a beautiful analysis of
the way in which the excess growth-ratio falls away to normal
as regeneration proceeds. The close approximation of his
results (p. 51) to the laws governing the flow of E.M.F.
between regions of differing electric potential is a further
165
i66
PROBLEMS OF RELATIVE GROWTH
justification for the provisional use at least of the term
' growth-potential '.
One peculiar fact demands notice. We have already seen
that the chela of the Gulfweed crab, Portunus sayi, affords
a good example of constant differential growth-ratio. Zeleny
(1905, analysed in Huxley, 1931B), however, also carried out
regeneration experiments ; and these are especially suited to
57/0
carapace length, mm.
15
Fig. 77. — Graph to show (upper curve) simple heterogony of the large claw
in the Gulfweed Crab, Portunus sayi, and (lower curve) the relation of the
amount regenerated during one instar to normal size of the claw (logarithmic
plotting) .
(Constructed from the data of Zeleny, 1905.)
our purpose since (a) he found that moult-period, not time,
was the essential factor affecting amount of growth in re-
generation, and (6) he always amputated a limb immediately
after one moult, and measured it immediately after the
next.
When his figures, both for the size of normal claws and for
PHYSIOLOGY OF GROWTH-GRADIENTS 167
those that had thus been regenerating for one instar, are plotted
on a double logarithmic grid against body-size, an interesting
result is obtained. Both sets of points approximate nicely
to straight lines ; but the line for the regenerates is inclined
at a slightly higher angle than that for the normal claws —
k = 1-20 instead of 1*15. In other words, during the first instar
after operation, large crabs regenerate a slightly greater frac-
tion of their claws than do small ones, and the absolute amount
of increase is multiplied by a constant factor for each unit of
multiplicative increase in absolute size. The amount of this
factor is, of course, obtained by dividing the ' growth-ratio '
determined from the slope of the line for the regenerating
claws by the actual growth-ratio as found for the normal
claws. This paradoxical result may possibly be accounted
for on the principle, which holds in many cases, that the
rate of regeneration increases with the amount removed.
We have already seen (p. 52) that when a heterogenic
organ is grafted on to a body relatively too large for it, its
growth-rate is decreased, so that here too it tends towards
its proper relative size. Thus here also the idea of a partition-
coefficient, representing an equilibrium between the amount
of material in the organ and in the body, is supported.
§ 2. The Initial Determination and Physiological
Basis of Growth-gradients
An important question is that of the physiological and
biochemical bases of heterogony and of growth-gradients.
Some experiments of Morgan (1932A) on male fiddler-crabs
give some indications on this problem. His results may be
summarized as follows. Young male fiddler-crabs, at a very
early instar of their post-larval existence, produce two claws
of male type. If one of these be amputated, the other is very
shortly afterwards fixed as permanently male-type, and will
regenerate male-type if later amputated. The amputated
claw regenerates of female type and is from thenceforth fixed
in this condition. Normally, it is the loss of one claw during
the first few instars which determines the right- or left-
handedness of the males.
If both claws are cut off during this early symmetrical stage
when both are of male-type, both regenerate of female type,
and remain so permanently. Occasionally, however, a male
is found in nature which has lost neither of its claws during
i68
PROBLEMS OF RELATIVE GROWTH
youth, and has grown to a considerable size while still the
possessor of two symmetrical male-type claws.1
When either of these is amputated, it regenerates male-type.
Thus we have the remarkable fact that while in early youth
the amputation of both male-type claws is followed by a loss
of all the male-type potentialities, and the amputation of one
by a loss of male-type potentialities on that side of the body,
this does not hold when the symmetrical double-male-clawed
stage has lasted to a considerably later period of life.
One possibility that suggests itself is as follows. We know
that during the phase of chemo-differentiation, prospective
potencies are sharply localized.
It is reasonable to suppose that
the potency for growing into a
large instead of into a small
chela is localized in the claw's
growth-centre, viz. the propus.
Whatever the chemical substances
responsible, they are then wholly
removed by amputation, just as
those responsible for limb-differ-
entiation in Amphibia are wholly
removed if a particular disc of
material be removed during em-
bryonic life. But we have seen
that in the male-type chelae of
Uca there is a growth-gradient.
If, as a result of this during growth,
the substance determining heter-
ogony and masculine type should
spread proximally to the breaking-
joint, then amputation should now permit the regeneration
of a male-type chela.2
1 In the only specimen of this type which I came across, the two
male claws together weighed more than the mean for a normal single
male-type claw for that body-weight, but considerably less than the
sum of two normal single claws. The range of individual variation
being considerable, however, one should have a number of specimens
before attempting to generalize.
2 In this connexion, reference should also be made to the results
obtained by Haseman (1907A and b) on the direction of differentiation
in segmenting Crustacean appendages. He finds that some regenerate
basipetally, others centripetally ; further, there is sometimes (e.g. in
many antennae), but not always, a particular segment which produces
Fig. 78. — Sketch to show asym-
metry in the thoracic portion
of the central nervous system
of the male fiddler-crab, Uca
piiguax. The shaded regions are
the ganglia supplying the chelae.
PHYSIOLOGY OF GROWTH-GRADIENTS 169
Another alternative is to suppose that the nervous system
is implicated. This is suggested by unpublished observations
of F. N. Ratcliffe on the nervous system of male Uca. He
rinds a marked asymmetry in the thoracic ganglionic mass,
naturally largest in the ganglion of the chelar segment but
visible in other segments as well. There are two classes of
cells in the nuclei associated with the ganglia, large and small.
The large are usually of different sizes but the same in
number on the two sides of the body. But the small, at
least in the ventral nucleus, appear to be equally distributed
in number in early stages and unequally distributed (larger
number on the side of the large chela) in later stages. Ratcliffe
tentatively suggests that this condition is brought about by
the permanent transfer to the side of the large chela of a
certain number of originally median cells which could be
transferred to either side of the body. If this suggestion be
substantiated (admittedly it needs further work for its verifi-
cation), it would imply that the normal fixing of the male-
type potentiality would be finally due to a certain number
of ' neutral ' nerve-cells being transferred to that side, while
the capacity of both claws to regenerate of male type in the
doubly male-clawed older males would be due to a loss, with
age, of the capacity of the median cells to transfer themselves
from one side to the other.
This suggestion would obviate the need of postulating the
proximal spread of a specific growth-promoting capacity in
the limb. It is, however, perfectly compatible with the idea
of an initial chemo-differentiative localization of high growth-
potential in the male chela.
In this connexion also the observations of Perkins (1929)
are interesting. In crabs (Carcinus, Cancer) and lobsters
(Homarus) he finds a gradient in the body as regards the con-
tent of glutathione, sulphydryl and other reducing compounds
known to be associated with growth. And this runs parallel
with the actual growth-relations of the various appendages
and regions of the body. He further advances a biochemical
hypothesis to explain the existence of growth-gradients ; but
as this is highly speculative, and as I do not pretend to
the other segments by repeated fission. In certain cases (1907B) he
was able to show that the normal directions of regenerative differen-
tiation could be reversed by special conditions. It is probable that
these facts are to be in some way related with those of growth-gradients,
but for the moment the connexion remains obscure.
170
PROBLEMS OF RELATIVE GROWTH
specialized biochemical knowledge, I will merely refer the
reader to his article (see Fig. 79).
It is noteworthy that in other respects he has found an
association of sulphydryl with growth-potential : e.g. the
decline with age both of sulphydryl content and growth-rate
(found in Carcinus, Pandalus, Sacculina embryos and Peri-
planeta). The coincidence of a gradient in this important
' key ' metabolic agent with observed growth-gradients is
Fig. 79. — Gradients in content of various sulphur compounds and in oxygen
uptake in the Crustacea, Cancer and Homarus (above), and the earthworm
(below) .
obviously a fact of considerable interest. Interestingly enough,
the gradient in sulphydryl content does not coincide with the
gradient in oxygen metabolism (in earthworms), so that Per-
kins concludes that the total oxygen uptake, being concerned
more with katabolic than anabolic processes, is not a good
measure of any gradients primarily concerned with growth,
a fact which clearly has important bearings upon Child's work
on the metabolic basis of his axial gradients.
§ 3. Other Gradient Theories
This brings us face to face with the relation between the
growth-gradients here described and other types of gradient-
effect. The existence of such effects has been casually recorded
OTHER GRADIENT THEORIES 171
on a number of occasions, and their importance for morpho-
genesis has been emphasized by such workers as Boveri, von
Ubisch, D'Arcy Thompson, and notably Child,1 who has
systematized the theory more thoroughly than other workers.
The more general conception of the morphogenetic field, of
which I believe gradient phenomena to be a particular case,
has been analysed from somewhat different points of view by
Weiss (1926), Guyenot (see Guyenot and Ponse, 1930), Hirsch
(1931) and Bertalanffy (1928). This is not the place to enter
into a general discussion of the subject, and I propose merely
to refer to a few relevant facts and ideas.
In the first place, two sets of essentially morphological facts
concerning gradients have long been known and recognized.
The first is generally subsumed under the title of the law
of antero-posterior development. It points out that during
development differentiation begins anteriorly and gradually
spreads posteriorly. Often the development of the head is
far advanced when that of the hinder end is not yet begun.
In some forms the undifferentiated posterior region may per-
sist throughout life ; or, as in many Crustacea, it may persist
through a considerable phase of free-swimming existence
although lost in the adult. In addition, there exists in bilater-
ally symmetrical animals a similar gradient in time of develop-
ment between dorsal and ventral surface. In Vertebrates the
region which leads the way is the dorsal mid-line, in Inverte-
brates in general the ventral mid-line. Subsidiary graded
effects of similar nature also occur within the appendages.
Secondly, a gradient also almost invariably occurs within
the ovum, along the main axis. This may be revealed in the
stratification of yolk or other materials, or in the greater rate
of segmentation at the animal pole, or in both ways. It is
the merit of Child that he has linked up these two sets of
facts in one general physiological theory. He has further
shown that the physiological gradient effects of which these
1 I prefer not to use the term metabolic gradients, also sometimes
used by Child. Child has not conclusively demonstrated that his
gradients are fundamentally metabolic in character ; but he has
demonstrated that, as regards morphogenesis, gradient-systems do
exist and are operative. The most important facts about these mor-
phogenetic gradient-systems are (a) that they are field systems in
which all the parts are interdependent within one plastic system, and
(b) that in some respects at least they are quantitatively graded.
Personally I would prefer the phrase morphogenetic gradient-fields, but
there is no need at the moment to complicate terminology thus. For
an excellent discussion, see Needham, 1931, p. 582 seq.
172 PROBLEMS OF RELATIVE GROWTH
are particular morphological expressions may continue through-
out life. He and his pupils have most thoroughly demon-
strated this for hydroid polyps and planarian worms. By this
work, certain important empirical laws have been established,
notably the fact that the first region to differentiate in re-
generation normally acts as a ' dominant ' region which has
a morphogenetic effect on the regions which differentiate later.
This has now been brought into line with the facts concern-
ing Spemann's ' organizer ' in Amphibian development (see
Santos, 1929). Further, such experiments as Stockard's cele-
brated production of cyclopia in Fundulus, cannot be inter-
preted except in terms of axial gradients. Many of Child's
empirical facts have been independently conformed by Abeloos
(1930), and the existence of a dominant region with morpho-
genetic effect by various workers, of whom Berrill (1931) is
the latest. (See also the work of Buchanan, p. 260.)
In regard especially to hydroids and planarians, Child has
been able to show that the physiological gradients constitute a
true field system, e.g in regenerating fragments of Planaria the
gradient can be either steepened (e.g. by optimum tempera-
ture) or flattened (e.g. by cold or by narcotics) : and when
this is done, it is found that the dominant region whose activity
has been depressed induces smaller dependent organs, at a
smaller distance than normal from itself, while the converse
holds when it has been heightened. Analogous experiments
have been performed on developing eggs, by using stimulatory
or depressant drugs, or by applying temperature-gradients.
The continuance of the gradient throughout life is shown
in many forms by their graded capacity for regeneration, e.g.
the head-frequency often decreases steadily in an antero-
posterior direction. There exists also a graded susceptibility
to poisons along the main axis. In forms which bud or divide
by transverse fission, the distance between the dominant
regions of old and new zooids appears to be determined by
the extent and steepness of the gradient. The production of
axial heteromorphosis in regeneration, such as biaxial heads
or tails, can also be satisfactorily interpreted in terms of the
gradient hypothesis. Child has attempted to explain these
facts in relation to differences in metabolic rate, but the proof
cannot yet be said to be conclusive. However, whether or
no this metabolic interpretation be correct, a set of impor-
tant empirical principles remain — notably that physiological
gradients exist in early stages of development, that they may
persist throughout life, that their slope and extent are deter-
OTHER GRADIENT THEORIES 173
mined with reference to a first-formed dominant region which
also has a morphogenetic or organizing effect, and that the
parts and organs involved in an active gradient system are
bound up together in a single physiological field, so that
alterations in one part will necessitate correlative alterations
elsewhere.
In connexion with the persistence of gradient effects through-
out life, we may refer to some further examples not cited by
Child. Apart from the continuous growth-gradients in single
organs, in body-regions and in the body as a whole which
we have discussed in this book, we have also discontinuous
growth-effects which we can only understand on the basis of
continuous underlying gradients (p. 152). And we have also
gradients affecting the rate of growth or regeneration of
epidermal structures such as feathers and hair (p. 100).
Clausen (1929), by grafting methods, has shown the exist-
ence in the tadpole tail of a gradient in susceptibility to the
autolysing agencies which operate at metamorphosis : skin
and muscle grafts from the anterior regions of the tail when
transplanted to the back undergo more rapid histolysis during
metamorphosis than do similar grafts from more posterior
situations. This is a significant fact, for it cannot well have
any particular functional or adaptive significance in relation
to the metamorphic process, and the difference, like that be-
tween the two horns of the rhinoceros, merely acts as an
indicator for the existence of some fundamental inherent
property of the organism.
To take quite another example, Alverdes has demonstrated
the existence of graded peristaltic activity in regions of the
mammalian gut, showing that physiological gradients may exist
even where no morphological differentiation is visible.
A gradient as regards regeneration is also seen in the results
of von Ubisch. From his data (1915) on regeneration of limbs
in the insect larva Cloe diptera, we can calculate that the
length of the femur of regenerated limbs after one moult-
interval is 44-2 per cent, of the length of the normal femur
for the fore-limb, 40-6 per cent, for the middle limb, and
367 per cent, for the hind limb. This differs in sign from that
for growth in Sphodromantis (p. 135), but the data are not
wholly comparable.
In a later paper (1922) he refers to the fact that Przibram
(1919) and Krizenecki (1917) obtained a gradient of opposite
sign for the regeneration of the limbs of mantids and meal-
worms and thinks that his work on Cloe might have given
174 PROBLEMS OF RELATIVE GROWTH
different results if extended over a longer period of regenera-
tion. But he adduces new and important results from grafting
experiments on earthworms, showing that a difference in
' differentiation potential', as when a young head is grafted on
an old body, gives much better results than the converse
experiment or than even autoplastic grafting of young heads
on young bodies. From this he deduces the existence of a
' differentiation gradient ' of importance in morphogenesis and
regeneration.
Sinnott (1930) has elicited a curious fact in regard to the
variability of cell-size in tissues of the petiole of maple-leaves
(Acer). He finds that there exists a gradient between surface
and exterior, variability being least in the size of epidermal
cells, and increasing cell-layer by cell-layer towards the centre
of the petiole.
We have also the well-known work of Lund (e.g. 1923,
1928) who finds in various organisms a system of gradients
in bio-electric potential persisting throughout life, and has
shown that organic polarity and morphogenesis can be con-
trolled by electrical means. A correlation between electrical
and metabolic gradients has been shown by Purdy and Sheard
(p. 260). And on the chemical side, the recent paper of Wata-
nabe (1931) may be consulted ; he finds a gradient in amount
of oxidizable substance in earthworms, running parallel with
the gradients postulated by Child for this organism.
It would doubtless be possible to multiply examples ; but
this deliberately heterogeneous list will serve to emphasize
the wide range of gradient phenomena which may exist in
the adult animal body.
It next falls to discuss the relations between the growth-gradi-
ents with which we have been concerned, and other gradient-
systems, notably the axial gradients, as I shall term them.
Here we are on speculative ground ; but there are certain
indications which make us suspect some real connexion be-
tween the two. In the first place, there is the co-existence
of a growth-gradient and an axial gradient in Planaria (see
Abeloos, 1928). It would be extremely interesting to follow
this up in greater detail and especially to see whether changes
in steepness of the axial gradient were quantitatively associ-
ated with changes in the growth-gradient. In this case, it is
worth recalling, size is the chief index of physiological age,
and growth-partition is almost entirely a matter of size,
whether in fed specimens which are increasing, or in starved
specimens which are diminishing in size.
OTHER GRADIENT THEORIES 175
It is important to note that in Planarians the morpho-
genetic effects associated with the gradient (induction of
pharynx and other organs by the dominant region) take place
early, but do not interfere with the persistence of physiological
effects of the gradient (e.g. regenerative capacity, growth-
intensity) throughout life. In a not dissimilar way, the realiza-
tion of normal morphological differentiation of a limb or tail
in Triton does not interfere with the persistence of the morpho-
genetic potency of the surrounding area to produce a new or
additional differentiation of the same type throughout life
(Guyenot and Ponse, 1930).
The gradient, though persisting throughout life, might be
altered in shape. It is more natural on various grounds we
would expect that it was more likely to be flattened than
steepened with age. If it were flattened, we should expect
that the potency of differentiation at a given body-level would
be altered to a potency originally characteristic of a more
posterior body-level. This is what appears actually to occur
in serially heteromorphic regeneration in Crustacea and insects.
The regenerated heteromorphic appendage almost invariably
is of a type which normally belongs to a more posterior seg-
ment— e.g. antenna regenerated in place of eye-stalk (Palae-
mon), or fore-leg in place of antenna (stick-insects.)1
For these and various reasons we may regard it as probable
that the primary axial gradient of the egg and early embryo
will normally persist, although doubtless often in somewhat
altered form, in later periods and probably throughout life,
even when we have no ready means, such as antero-posterior
differentiation, heteromorphic regeneration, or head-frequency
in regeneration, of deducing its existence.
Further, if it does persist, we may again regard it as prob-
able, especially in view of the facts in Planaria, that it will
exert some influence upon growth. This influence may be
direct or indirect, but in any case would be graded in its effect.
Further, to reverse the approach, we may as a matter of
speculation conclude that it is probable, when gradients
specifically concerned with growth, like those in the male
chelae or female abdomena of Crustacea, are found, that these
1 Recent work by Przibram (Akad. Wiss., Vienna, 9. vii. 1931) and
Suster (ibid.) confirm this view. In Dixippus, antennae amputated
in the 1st instar regenerate as antennae, while in later instars they
regenerate leg-like organs. In Sphodromantis, the regenerate forms
an antenna at 25°, but a leg-like organ at lower temperatures. In
both cases the leg-like organ is produced when the axial gradient may
be presumed to be flattened.
176 PROBLEMS OF RELATIVE GROWTH
have elements in common with the axial gradients of Child —
viz. a dominant region (here the growth-centre) which exerts
a graded effect on neighbouring regions, so that the whole
system is a field-system.
It is already clear that in Child's scheme, place will have
to be found for qualitative differences among gradients. The
primary morphogenetic gradient in, e.g., a worm must differ
qualitatively from the activity-gradient concerned with the
addition of new segments throughout life. The primary
animal-vegetative axial gradient in an Amphibian egg must
differ qualitatively from the latter field-system of which the
organizer (dorsal lip) is the ' dominant region '. And specific
growth-gradients will constitute another main type.
This section has been admittedly very speculative, but as
tentative conclusion we may suggest that gradient-systems,
all perhaps of essentially similar nature, are concerned with
primary differentiation, the time-relations of early develop-
ment, certain physiological properties of parts of the adult
organism, certain regenerative capacities, and with growth-
intensity. These gradient-systems will all obviously in the
long run be ' metabolic ', but may be specialized in qualita-
tively different ways according to the type of activity which
is graded within them.
The primary gradient of the egg and early embryo may
be expected to persist throughout life and to have a minor
effect on the graded distribution of growth-promoting sub-
stances— i.e. some growth-gradients will be secondary effects
of the primary axial gradient. But in addition we may expect
that gradients concerned specifically with growth-intensity
may come into existence supplementary to and largely inde-
pendently of the primary axial gradient, but will then exert
their indirect effect upon such proportions of the primary
gradient as still persist.1
§ 4. Heterogony and Hormones 2
We must also consider the relation of hormones to hetero-
gony. Here we must at the outset remind ourselves of an
important point — that the action of a hormone always demands
1 The important paper of Smirnov and Zhelochovtsev (1931) has
appeared too late to receive the discussion it merits. It contains a
detailed mathematical analysis of relative growth in the leaves of the
Nasturtium (Tropaeolum major) under conditions of normal and
reduced illumination and brings the results into relation with a general
conception of a ' gradient-field ' of growth. See also Werner (p. 258) .
2 See also the work of Robb (p. 257).
HETEROGONY AND HORMONES 177
two specificities — the specificity of the hormone, and the
specificity of the tissue which reacts to the hormone. For
instance, there may or may not be a hormone concerned in
the growth of the fiddler-crab's large chela. Should there be
one, however, its action would in this case be subordinate to
the action of the specific capacity for heterogonic growth
possessed by the male type but not by the female type of
chela, whereas in a case like that of the fowl's comb, the
tissue-specificity is apparently the same (or almost so) in both
sexes, and the sexual differences in comb-size are brought
about by the specificity of the two sex-hormones.
So far as we know, there exist no sex-hormones in insects.
Accordingly, in this group any secondary sexual heterogonic
organs will depend for their development entirely upon their
inherent growth-capacities, which differ in the tissues of the
two sexes according to the cellular metabolism induced by
one or other sex-chromosome complex. In vertebrates, how-
ever, the reverse is usually the case : the tissue-capacity is
the same or highly similar in both sexes, and the sexual dif-
ferences are due to differences in sex-hormones. A good deal
of work has been done on the growth of the fowl's comb by
Pezard, Benoit, Lipschutz and others (references in Gold-
schmidt, 1923) . It appears that at the onset of sexual maturity
in males the growth of the comb becomes highly heterogonic,
and approximates to a constant differential growth-ratio. The
growth of the comb in females is also heterogonic, but mildly
so ; while in castrates it is isogonic. The marked comb-
changes associated with the onset of a laying period in a
pullet may be associated with a change in sex-hormones, or
quite possibly with a change in general metabolism. Benoit
(1927B) has shown that the growth-coefficient of the comb in
growing fowls differs according to the season at which they
are hatched. In those hatched in March- June, the coefficient
is high early, then decreases markedly in late summer, to
resume its high level in October or November ; in those
hatched after June, there is no slackening, but the initial
coefficient is lower. Benoit suggests that all the phenomena
are due to seasonal variations in testis activity. It is prob-
able that besides the sex-hormones, many other factors, such as
nutrition, influence the growth-partition coefficient of the comb.
Castration in adult males is followed by a regression in
comb-size ; this takes place according to a well-defined
mathematical formula. As set forth by Pezard (1921) the
12
178 PROBLEMS OF RELATIVE GROWTH
comb's regression-curve is parabolic, being represented by
the formula
L=1 + |C(0+O2
where L is the length of the comb after the lapse of time t
from the onset of regression, I the final length at the end of
regression, 6 the total time taken for regression, and C a
constant (varying from individual to individual). This is of
some interest, as body-size does not enter into the formula
at all. The converse curve, of comb-growth produced by
injection of male sex-hormones in capons has been recently
determined by Blyth, Dodds and Gallimore (1931). The
authors do not discuss this aspect of their work, but plotting
their data shows that the curves for re-growth are quite
different from Pezard's curves for regression. They show a
well-marked point of inflexion and are often very regular.
They could be represented by an expression of the form
x = A + B#2 — Cx3, which is equivalent to saying that
Robertson's autocatalytic growth-formula would apply to
them. It is interesting to find this difference between the
positive and negative aspects of the same growth-process.
Grafts of female comb or wattles on to male hosts become
larger than similar grafts on hosts of their own sex (Kozelka,
1930). The response of the comb to the sex-hormone is
brought about by a specific mucoid layer in the dermis
(Hardesty, 1931) : i.e. this is the true heterogonic tissue.
A great deal has been written as to the ' all-or-none ' law
of the action of the sex-hormones on comb-growth, some
writers, like Pezard, maintaining that it holds, others, like
Benoit, opposing the idea. It would appear that Benoit is
correct, but that the range over which the action of the sex-
hormones is proportional to its amount is very limited. Above
this point, the maximum reactivity of the comb has been
reached, and increase of testis-size is not followed by further
increase of comb-size.
That the matter need not be so simple as this, however,
is shown by experiments on mammals. Collip (1930), by in-
jecting a particular fraction of placenta-extract into rats, has
produced in both sexes accessory sexual organs (seminal
vesicles, vagina, etc.) far exceeding in absolute and relative
size anything normally found in the species ; on the other
hand, injection of anterior pituitary (in females) causes an
acceleration of maturity but without disproportionate size of
HETEROGONY AND HORMONES 179
the accessory organs relative to the gonad (see references in
Parkes, 1929, pp. 158-9). Thus it would seem that pre-
pituitary extract causes acceleration of the growth of the
gonad and accessory organs, but without inducing an increase
in the final maximum size of either, or a disproportionate
development of accessory organs relative to gonads ; while
injection of placenta-extract leaves the gonad of normal rela-
tive size, but apparently induces a supernormal production
of sex-hormone, this in turn resulting in supernormal hyper-
trophy of accessory sexual organs. We may expect to find
similar complications elsewhere as regards the relation of
hormone-producing organ and reactive organ.
Champy (1924) has given us an interesting experiment on the
dorsal crest of male newts (Triton). This is a male secondary
sexual organ, appearing at the onset of the breeding season,
and exhibiting marked heterogony. Champy finds that it
shows a differential response to starvation, being reduced
relatively faster than other organs.1 This may be due to the
crest consisting of material which is readily drawn upon in
starvation, or may point to some more general law of the
reversibility of heterogonic growth, according to which a
heterogonic organ would always tend to approach the size
appropriate to its partition-equilibrium whatever the bulk of
the body, and whether that bulk was being increased by normal
growth or reduced by starvation. That something of this
latter sort may occur is shown by the well-known fact that
planarian worms reduced in size by starvation revert to juvenile
proportions (Child, 1915). This relation has recently been
worked out quantitatively by Abeloos (1. c), who finds not
only that the change of proportions during normal growth is
truly heterogonic and approximates to that obtainable by
constant differential growth-ratio, but that it is almost exactly
quantitatively reversed during reduction due to starvation.
In any case, the example of Triton clearly demonstrates the
co-operation of nutritive and hormonic factors in determining
the size of an organ (Fig. 80).
Non-sexual organs may, of course, also show heterogony,
and their heterogony may also depend upon hormones. This
is best seen as regards the growth of limbs in Anuran meta-
morphosis, which depends upon thyroid hormone, and is, up
1 Unfortunately no weight-measurements of crest and other soft
parts were made ; but Champy's illustrations appear in general to
bear out his assertion.
iSo
PROBLEMS OF RELATIVE GROWTH
to a considerable dosage, above which no further increase of
reaction is obtainable, a function of the amount of hormone
administered. Champy (1922), as a result of investigations
on thyroid-fed frog tadpoles, concludes that thyroid, during
Fig. 80. — To show disproportionate reduction by starvation of a heterogonic
organ (male dorsal crest) in a male newt, Triton cristatus. The drawings are
to scale, and were made at o, 5, 9, 16, 24, 32, and 45 days from the beginning
of starvation.
HETEROGONY AND HORMONES 181
the few days between its administration and metamorphosis,
causes a progressive geometric increase in the number of
dividing cells. This would imply a progressive increase in
growth-ratio during the period. Further researches on this
interesting point are needed.
In any case, the legs of Anura are of interest to us in several
respects. First of all, they emphasize once more the import-
ance of tissue-specificity. While their growth responds very
sensitively to thyroid, the larval Urodele limb is wholly in-
sensitive to the same hormone. Next, they show us that
even the threshold of reactivity to one and the same hormone
may vary from tissue to tissue. Anuran limb-buds appear to
be sensitive to any dose of thyroid, starting from zero. But
the equally specific degenerative response of the tail-tissues
does not begin until a considerable concentration of thyroid-
hormone is reached in the blood. Thirdly, they show us a
clear-cut case of the growth-coefficient of an organ varying
within wide limits with the dosage of a hormone. Even the
normal growth of the Anuran tadpole's limbs is slightly hetero-
genic, as is shown by extirpating the thyroid in the embryo,
when growth is much lower, and apparently isogonic (Allen).
Fourthly, they show in diagrammatic form the interrelation
of the factors of growth-coefficient and available time as regards
the problem of relative size of organ. Normally, the slight
leakage of thyroid hormone into the blood during larval life
produces a mild limb-heterogony. By the time the meta-
morphic crisis occurs (apparently due to a specific change in
the pituitary which causes the thyroid to liberate most or all
of its secretion suddenly into the blood), the hind-legs, originally
mere buds, have had time to increase in relative size until
longer than the trunk, although the trunk itself has increased
perhaps two- or three-fold in linear size.
If, on the other hand, a moderately strong dose of thyroid
be administered to small or small-medium tadpoles, with limb-
buds hemispherical or conical but not yet fully differentiated,
then although the growth-ratio of the limb-bud is raised far
above normal, the accelerated growth can only operate for a
few days before metamorphosis supervenes, with the result
that the transformed froglet has relatively very small limbs.
It would be of great interest to see whether these proportions
were later regulated to or towards the normal, but the experi-
ment has not so far been tried. As suggested in Chapter II,
this factor of the amount of time available for heterogonic
182 PROBLEMS OF RELATIVE GROWTH
growth appears to be of importance in holometabolous in-
sects in general, and in the dimorphism of male Forficula in
particular. (See Fig. 81.)
Finally, anuran limbs well illustrate the dependence of
differentiation upon growth. It is often asserted that in
Amphibia the thyroid hormone ' favours differentiation but
inhibits growth '. This is inaccurate and misleading. Slight
thyroid activity, as in the normal tadpole, is not incompatible
with total weight-increase. Excess thyroid causes loss in total
weight, but this is an effect on balance, many organs losing
weight, others, like the limbs and skeleton, gaining weight.
As Champy (1925) has clearly shown, the effect of thyroid on
some tissues is to halt their growth (gut) or even to cause
their atrophy (tail, gills), on others is neutral,
and on still others is to increase their
growth (limbs). Further, the differentia-
tion of the limbs is not a specific effect of
the thyroid hormone, but a secondary effect
of their growth in size. Differentiation of
limb-segments, digits, etc., occurs at certain
limb-sizes. And it will do so even when
Fig. 81. Dispro- no thyroid hormone is present, as is shown
portionately small by Allen's thyroidectomized tadpoles (Allen,
limbs caused by jqjg iqiq). These grew to a giant size
precociously induced /r , 1 \ a • t 1
metamorphosis in (for tadpoles). As result, their limbs,
the common frog, though growing isogonically, attained an
JengthSof°the "frogiet absolute size comparable to that reached
was 8 mm. by the limbs of normal tadpoles a few
weeks before metamorphosis ; and they
showed a comparable degree of differentiation. This depen-
dence of type of differentiation upon absolute size of organ
is frequently to be met with (cf. in prawns with different
heterogony of the chelae in both sexes, the resemblance of
the proportions of heterogenic male and female chelae of the
same absolute size, but of very different ages and attached
to bodies of very different absolute size : see Chapter III).1
In a later chapter we shall see that it has important taxonomic
and evolutionary consequences.
The work of Hutt (1929) shows an interesting effect of the
male sex-hormone upon the proportion of limb bones in
1 It is however not invariable. Male and female chelae of Maia,
Uca, etc., grow according to quite different growth-gradients (Chap-
ter III).
HETEROGONY AND HORMONES 183
fowls. The annexed table, modified from his Table 5, shows
the chief results.
TABLE XIIa
Percentage Change in Size in Parts of the Limbs of Male Fowls
Induced by Castration
Carpo- Phalanges of
Humerus Radius, Ulna metacarpus digit 3 (mean)
Fore-limb . + 2-6 +2-25 +2-2 — o-i per cent
Hind-limb . +3-0 +3-4 +3-9 +4 -7 per cent
Femur Tibio- Tarso- Phalanges of
tarsus metatarsus digit 3 (mean)
It will be seen that castration causes in general an increase
in the size of the limb-bones. But whereas in the hind-limb
the increase itself increases as we pass distally, in the fore-
limb it is graded in the reverse sense, leading to an actual de-
crease in the terminal segment. The existence of the gradients
is interesting, but the explanation of their opposite sign in
fore and hind limbs is at present quite obscure.
Hammett (1929B) has published a summary of our knowledge
of the effect of thyroid upon growth. In the first place, the
effect upon total growth is an affair of dosage. In intact
young mammals (and probably many other vertebrates) slight
excess of thyroid causes an increase of growth in weight, while
heavier doses, by a differential encouragement of katabolism,
reduce it. In thyroid-deficient animals (whose growth is
usually retarded), much greater doses will, of course, still
permit increased growth.1
Our chief interest, however, concerns the differential effect
of thyroid activity upon bodily proportions, and here Ham-
mett himself has made elaborate experiments upon albino
rats. Groups of these were thyroidectomized at 23, 30, 50,
65, 75 and 100 days respectively, and their organ sizes and
weights determined and compared with those of unoperated
controls at 150 days. When the increments made by the
various organs measured in the operated animals are calculated
as percentage of the increments made in the same space of
time by the same organs of the controls, some important facts
emerge. First, in every case the effect of thyroidectomy in
retarding growth (not the absolute effect, but the relative effect,
1 Thyroidectomy in some animals (anuran tadpoles, axolotls, etc.)
is not accompanied by any change in growth-rate. In anuran larvae,
excess thyroid is usually accompanied by a decreased growth (incipient
metamorphosis). Even here, however, very minute doses accelerate
growth during the early part of the pre-metamorphic period.
184 PROBLEMS OF RELATIVE GROWTH
as measured in the way described) increases with the age of
the animal. To take but a few examples, the eye, one of the
organs least affected by thyroidectomy, in males operated at
twenty-three days, by 150 days had made nearly 100 per cent,
of the growth of the eyes of the controls in the same period.
For those operated at 100 days, however, it made less than
80 per cent, of the controls' growth between 100 and 150
days.
In regard to total body-weight, males operated at twenty-
three days showed about 60 per cent, of the increment of the
controls, while for those operated at 100 days, the percentage
was below 30 per cent. And as regards kidneys, males operated
at twenty-three days showed under 50 per cent, of the con-
trols' increment, those operated at sixty-five days showed hardly
any increase at all, while those operated at 100 days, had
before 150 days lost in absolute kidney-weight an imount
equivalent to some 50 per cent, of the increment made by
the controls in the same period. Thus the sensitivity of
growth to thyroid-deficiency increases with age.
Secondly, most of the organs of the body fall into distinct
groups as regards differential sensitivity to thyroidectomy.
The eye-balls, central nervous system, length-growth both of
body and tail (doubtless determined by growth of the axial
skeleton), and both length- and weight-growth of humerus and
of femur are relatively resistant, being retarded less than the
weight of the body as a whole. On the contrary, the adrenals,
spleen, kidneys, liver, heart, submaxillary glands, pancreas
and to a slighter extent the lungs, make less increment than
general body-weight (Fig. 82).
Hammett points out that these results are all consistent
with the idea that the influence of thyroidectomy is greater
on growth by increase of cell-size than on growth by cell-
multiplication ; greater on that fraction of the metabolism
concerned with function (work) — e.g. secretion, muscular
activity, etc. — than on that concerned with growth ; and
greater on labile than on stable chemical compounds. The
last point is a correlate of the obvious fact that in conditions
of subnormal nutrition (as in thyroidectomy) labile materials,
such as the contents of glandular tissues, are more readily
drawn upon than stable or inert substances, such as the salt-
deposits in the skeleton, the lipoids of the nervous system,
or the humours of the eye. The second is a correlate of the
fact that in malnutrition, growth gives way to the mainten-
Fig. 82. — Differential effect of thyroid removal upon the growth of various
organs in the male Albino rat.
The curves indicate the amount of growth made by an organ, represented as a percentage of the
amount of growth made by the same organ in unoperated controls. For each organ there are 6 points,
representing operations carried out at ages from 23 to 100 days ; all the 6 series were allowed to
grow until 150 days of age before final organ-size was determined. In all graphs the heavy line
represents the curve for body-weight in thyroidectomized animals. In all cases the effect, relative
to the control, is greater in animals operated later.
(A) Organs of resistant type : eyes, spinal cord, brain. All are less reduced than body-weight.
(B) Organs of sensitive type : heart, lungs. The heart in particular is more reduced than body-
weight. In animals operated at 100 days of age, both heart and lungs actually lose weight (negative
increment).
185
186 PROBLEMS OF RELATIVE GROWTH
ance of function, as shown by the fact that young mammals
can live fairly healthily when kept at constant weight or even,
for a time, decreasing weight, and that planarians, etc., can
keep healthy though enormously reduced in bulk by starva-
tion. In organs where, to use Hammett's phrase, the work-
growth ratio is high, as in glands, heart, etc., since a decrease
of metabolism falls more heavily on the growth-function than
the work-function, therefore the growth-function will be more
seriously impaired than in organs such as C.N.S. or skeleton,
where the work-growth ratio is low. And the first point, the
greater influence of thyroid-deficiency on growth by cell-size
increase, is perhaps due to a specific action of thyroid, though
it too could be interpreted in terms of a greater effect of thyroid-
depression on growth than on function or work, the cell which
is growing in bulk being usually already differentiated for its
definitive function.
Hammett also considers a ' special group ' of organs. The
thymus is noted for its sensitivity to all unfavourable con-
ditions ; and in general is the most adversely affected of all
organs by thyroid removal. The testes are resistant because
continuously producing new cells, while the ovaries, after
puberty at least, are highly affected. The pituitary is known
to show compensatory hypertrophy on thyroid removal, and in
consequence increases its growth, notably in males (Fig. 83).
There is also a specific sex difference, females showing a
greater effect of thyroidectomy as regards body-weight and
almost all organs. And finally, puberty accentuates the
sensitivity of all the sensitive organs.
The net result of thyroidectomy is the production of an
absolutely smaller 150-day animal, but one which is abnormally
slender, and has relatively larger eyes, testes and seminal
vesicles, nervous and skeletal systems than a control of the
same age, but relatively smaller viscera and glands ; further,
the normal sexual size-difference is increased. When compared
with animals of the same body- weight, the resistant groups of
organs are, of course, absolutely as well as relatively larger
(Hammett, 1929A).
Hammett further states that there is no correlation between
the degree of sensitivity to thyroidectomy and the normal
growth-rate (growth-coefficient) of organs. However, he has
for the most part compared qualitatively different organs, in
which the work-growth ratio, the cell-multiplication : cell-size-
increase ratio, and the ratio of labile to stabile materials may
HETEROGONY AND HORMONES
187
160
140
120
readily obscure differences due to inherent growth-potential.
To detect the latter, it would be necessary to compare the
effects on different members or regions of a single organ system
— e.g. different bones — which have different growth-ratios.
This interesting study at least shows the complexity of the
factors by which ' normal proportions ' are determined.
Here we may cite
the paper of Neva- male £
lonnyi and Podhrad-
sky (1930), which
does indicate a differ-
ential result of excess
thyroid (and of ex-
cess thymus) on vari-
ous parts of the
skeleton of fowls.
Unfortunately the
thyroid results are
based on only two
experimental and
four control animals,
so that confirmation
is required. It would
appear, however,
that on the whole
there is a tendency
for excess thyroid (in
this dosage) to pro-
duce relatively
thicker bones ; fur-
ther, to encourage
the total growth of
the femur, shoulder-
girdle and hip-girdle,
but to decrease the
1 e n g t h-growth at
least of the meta-
carpals and metatarsals ; the radius, ulna and tibia occupying an
intermediate position. This is evidence, so far as it goes, of
a graded centrifugal effect. We may also note that Hammett
found femur less sensitive to thyroid defect than humerus,
and the Czech authors found it respond more by excess growth
to thyroid excess.
Fig. 83. — Effect of thyroidectomy on the growth
of the hypophysis in the male Albino rat.
The graph is constructed as described in Fig. 82. The hypo-
physis is the only organ which increases more in hyroidec-
tomized than in Control animals.
188 PROBLEMS OF RELATIVE GROWTH
There is one point in Hammett's discussion which needs
further consideration. He maintains that all the growth-
effects of thyroidectomy are what we may call non-specific,
due to its differential effects on different general kinds of
materials or metabolic activities. That this need not always
be so, however, is conclusively shown by the facts of Amphibian
metamorphosis ; in Anuran larvae thyroidectomy has a much
greater growth-retarding effect on the limbs than on any other
organ, in spite of the fact that the growth of these, normally
and under the effect of excess thyroid, is mainly due to cell-
multiplication (see above). There is, in fact, the possibility
of specific as well as of general differential effects of hormones
upon growth.
This has been well brought out by Keith (1923), who has
pointed out that the pre-pituitary affects growth differentially,
acting most of all upon the extremities and on the parts con-
nected with jaw-function. And similar specific sensitivity is,
of course, abundantly shown in regard to the sex-hormones
(see above). Cushing (1912) points out that hyperpituitarism
is associated with relatively stout digits, hypopituitarism with
relatively slender ones.
Although there are no sex-hormones in insects, reference
may be made here to Champy's interesting discussion (1929,
pp. 204 seqq.) of the relation between heterogony and secondary
sexual characters. In various genera of beetles, in which a
horn or other excrescence is normally present in males only,
it occurs in both sexes in certain species. When this is so,
the organ appears almost invariably to be highly heterogonic
when restricted to one sex, only slightly heterogonic or even
isogonic when present in both sexes (e.g. the beetle Phanaeus
lancifer as against other species of the genus). In Enema
infundibulum, and other beetles, the female possesses a cephalic
horn, but lacks a thoracic horn : in the male, the latter is
highly heterogonic, the former only slightly so.
The same phenomenon is found as regards the ' tails ' of
the swallow-tail butterfly, Papilio dardanus. In the few sub-
species where the sexes are similar, the ' tail ' is isogonic ;
in the rest, it is heterogonic. He further states that the same
phenomenon occurs in Chameleons, Sheep, Antelopes, and
Deer (Reindeer, as against the other forms). Here, however,
quantitative data are lacking, and the statement needs
confirmation.
Champy correctly stresses the frequent relation between
HETEROCHELY 189
heterogony of a character and its restriction to one sex. It
will be clear, however, from our previous discussion that
heterogony is a quite general phenomenon, and that it is
merely very obvious in many secondary sexual characters be-
cause these are frequently of an exaggerated nature, demanding
high growth-ratios for their development.
§ 5. Heterochely and Relative Growth-rates
Some suggestive facts concerning differential growth emerge
from the experiments on heterochelous Crustacea, which have
been summarized and interestingly discussed by Przibram
(1930). As is well known, in forms like Alphaeus, while ampu-
tation of the small or nipper claw is followed simply by the
regeneration of a new nipper, with no change in the crusher,
amputation of the crusher is followed by a reversal of the
type of claw, the old nipper growing into the new crusher,
the claw which regenerates in place of the old crusher growing
into the new nipper. (This, it may be recalled, is closely
parallel with the results of Zeleny (1905) on the polychaete
Hydroides. This sedentary worm has paired opercula, of
which one is large and functional, the other rudimentary.
Amputation of the large operculum causes the rudiment to
grow into a functional organ, while the amputated organ
regenerates of rudimentary type.) In certain other hetero-
chelous Crustacea, such as the lobster (Homarus), it was long
supposed that amputation of the crusher was not followed by
this reversal. But later work has now shown that this only
applies to later stages : if the operation is performed on quite
young animals, reversal does occur. And it now seems to be
a general rule that reversal will occur at small sizes in all
heterochelous forms, although the size-range over which it
occurs differs considerably from form to form. In some cases
no reversal is possible after quite a young stage. In others,
like Alpheus, it can occur throughout life. In both Homarus,
in which the crusher (like the large chela of male Uca) may
be either on the right- or left-hand side at random, and in
the crabs Portunus and Eriphia, in which the crusher is nor-
mally always on the right, reversal can only take place before
visible differentiation of claws has occurred. The result of
amputation of future crusher in such cases is that the other
claw (as in young Uca, where, however, both chelae are similar,
of male type) becomes the crusher, while the amputated
claw regenerates as a nipper.
igo PROBLEMS OF RELATIVE GROWTH
Further, in practically all species, even in those where
reversal will take place after visible heterochely has appeared,
it will be either slowed down or totally prevented after the
attainment of a certain quite small body-size. Above a
cephalo-thorax length of about I cm., if the crusher be ampu-
tated, both chelae appear of nipper type at the next moult
following amputation of the crusher ; and in most such cases the
regenerating claw usually turns into the crusher again after one
or more further moults. (The only exceptions are hermit-
crabs, in which asymmetry of the whole body, and consequently
of the chelae, sets in during the larval development, and
appears to be invariably determined by the start of post-
larval life.)
Przibram ingeniously suggests that these facts are due to
the ratio between rate of regenerative and of normal growth
at the time of operation. Rate of regenerative growth is
always high. We have little information as to its decline
with total size, but the previously cited experiments of Zeleny
on Portunus sayi (1. c.) show that there need be no falling-off.
Probably rate of regenerative growth remains about constant
where regeneration is possible at all. Normal growth, on the
other hand, falls off very markedly with absolute size-increase.
Thus the growth-ratio of regenerating crusher to normally-
growing nipper will increase with increase of absolute size.
There is, further, some evidence which suggests that a smaller
organ (such as the nipper) can in the absence of competition
from a corresponding larger organ (such as the crusher) draw
on the blood for a greater amount of nutriment.
Przibram accordingly suggests that when the growth-ratio
of regenerating crusher to normal nipper is relatively low, the
impetus given to the nipper's growth by removal of the crusher
can reach the threshold of growth-coefficient needed to pro-
duce a crusher before the regenerate can catch up, upon which
the regenerate is then partially inhibited and must remain as
a nipper ; but when this growth-ratio is high, the regenerate
can differentiate as a crusher at the next instar, because the
amount of excess growth achieved by the nipper is relatively
so low. An intermediate condition is seen when at the next
instar both chelae regenerate as nippers, though the original
nipper is still the larger ; the nipper has not yet got sufficient
impetus to differentiate into a crusher at the first instar ;
and although the growth-ratio of the regenerate has not been
sufficiently high to reach crusher-type at this instar, its growth-
INTERACTION OF GROWTH-RATIOS
191
/f>C
r. ! n
r ! n
im! an
1
1
1
1
1
r ! N
(a)
(b)
(c)
ratio is still considerably higher than the old nipper, and at
the next moult it will almost inevitably overhaul its partner
(Fig. 84). This hypothesis fits the facts very prettily; it
only remains to confirm it quantitatively by appropriate ex-
periments. It should, however, be pointed out that in any
case it only explains why one of two alternatives is realized
on a given side of the body in each particular case : it gives
us little light on the ques-
tion of the nature and
determination of the two
alternative types them-
selves. The difference be-
tween crusher and nipper
is doubtless in part one of
absolute growth-coeffici-
ent ; but as we have earlier
seen, there are also differ-
ences between them in the
spatial distribution of
growth-potential, and
qualitative differences in
teeth, etc. The changing
growth-ratio between re-
generating and normal
claw acts primarily as a
' realization factor ', to use
a German term, just as
does the female sex-
hormone of birds in effect-
ing the difference in colour
and form between male and female feathers. The determina-
tion of the difference in growth-gradients and in teeth is not
in the least explained by this.
Fig. 84. — Diagram to illustrate Przibram's
hypothesis as to reversal of chelae in
heterochelous Crustacea.
C, crusher ; N, nipper claw.
In each case the crusher has been removed, and its
regeneration rate is assumed to remain constant in spite
of age. Dotted lines indicate growth. The height of
the lines is not supposed to indicate absolute size, but
is adjusted so that greater height indicates capacity to
develop into a crusher and to inhibit the other claw,
causing it to become a nipper.
(a) In young animals the nipper grows relatively fast
and can thus differentiate into a crusher at the next
moult, causing the regenerating crusher to become a
nipper ; [b) in old animals the nipper grows relatively
slowly ; the regenerating claw can thus differentiate
into a crusher again at the next moult, and the nipper
remains a nipper ; (c) at an intermediate stage, neither
claw gets sufficient advantage to develop into a crusher
by the first moult ; but at the succeeding moult the
old crusher will become a crusher again.
§ 6. Specific Growth-intensities and their Interaction
A series of highly important experiments bearing on our
problems has been carried out by Harrison (1924, 1929), and
later by Twitty and Schwind (1931), in grafting organs from one
species of urodele larva to another which has a different normal
growth-rate. The species used were Ambly stoma tigrinum,
which has a high growth-rate, and A . pimctatuni which, even
when fed as fully as possible, grows at a considerably lower
192
PROBLEMS OF RELATIVE GROWTH
rate in the same conditions ; and they then grafted limb-disks
and eye-rudiments from one species to the other. The main
important finding is that the organ retained their specific
growth-rates even on the bodies of their new hosts, provided
that conditions were equalized by keeping both species at the
same nutritive level (which was most simply accomplished by
giving both the maximum amount of food they would take).
Under these conditions, the transplanted organ attained the
same size which it would have had if left in the body of its own
species ; tigrinum organs grew just as fast as normal though
on the slow-growing bodies of punctatum, punctatum organs
just as slow as normal though on the fast-growing tigrinum
bodies (Fig. 85). On the other hand, this inherent growth-
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DAYS AFTER OPERATION
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Fig. 85. — Graph showing the growth of normal
(A) Growth of tigrinum eye grafted on to punctatum, compared with that of a normal tigrinum
eye, and that of the normal punctatum eye of the host.
INTERACTION OF GROWTH-RATIOS
193
capacity of the organ was related to the growth of the body (I
am now interpreting the data in terms of the views put forward
in this book) according to the law of constant coefficient of
growth-partition (Figs. 31, 86). In other words, the amount
of growth made by the organ was a function not only of its
own higher or lower growth-potential, but also of the amount
of growth made by the body, and this was equally true both
for normal and transplanted organs. For instance, at first it
appeared that tigrinum (rapid-growing) organs grew even
faster when transplanted than when left in place, and Harrison
put forward an elaborate hypothesis to account for this. It
later turned out, however, that this was due to the punctatum
hosts having been at an optimum nutritive level, while the
tigrinum controls were not being given quite as much food as
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DAYS AFTER OPERATION
B
and grafted eyes of two species of Amblystoma.
(B) Reciprocal experiment. Growth of punctatum eye grafted on to tigrinum, compared with that
of a normal punctatum eye, and that of the normal tigrinum host eye.
13
194
PROBLEMS OF RELATIVE GROWTH
or
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DAYS AFTER OPERATION
Fig. 86. — Similarity of relative growth of eye of two species of Amblystoma
with different absolute growth-rates.
The curves are derived from the means from 5 specimens of each species, which were used in eye-
grafting experiments. They represent the ratio between the diameter of the normal unoperated eye
and the total length.
The ratios decrease (negative heterogony) but are closely similar for the two species in spite of
the fact that during the period the mean-length of the tigrinum larvae had increased from about
14 to no mm., while that of the punctatum larve had only increased from about 13 to nearly 55 mm.
The operation was carried out in the embryonic period.
they could eat.1 By reversing the nutritive situation, the
amount of growth made by the grafted limbs could be reduced
1 The matter is actually not quite as simple as this. From the
detailed results of Twitty and Schwind (1931) it appears that the
correspondence between grafted and control limbs only remains perfect,
even in maximally fed specimens, up to a certain time in larval life.
After this, there is a slight decrease in the size of the large limbs grafted
on to the small host, and a more considerable increase of the small
limbs grafted on to the large host (Fig. 85) — an effect the reverse of
that first posited by Harrison. It would thus appear that the size
of the host gradually comes to influence the inherent growth-capacity of
the graft, although this influence is always slight compared to the differ-
ence in the inherent growth-capacity of the organs of the two species.
Further, Detwiler (1930) and Severinghaus (1930) claim that the
host may in some cases exert some effect upon the inherent growth-
rate of the graft in early stages before feeding begins. But what-
ever the extent of these minor effects, the specific growth-rate of the
organ is the main factor in determining its relative growth-rate when
transplanted.
INTERACTION OF GROWTH-RATIOS 195
below that made by the controls left in place. Thus we clearly
see that the relative size of any organ is due to two variables
— its inherent growth-capacity, and a regulatory partition-
coefficient which allots material as between organ and rest-
of-body.1 We could have deduced this from our studies on
the chelae of Uca, etc., but this work clinches the matter
decisively by means of experimental tests.
This work of Harrison's has also shed important light on
growth-gradients. In addition to grafting whole eyes, he made
separate inter-specific grafts (a) of optic vesicle and (b) of
lens-producing epithelium (Harrison, 19293). Eyes were thus
produced which contained rapid-growing optic cup and slow-
growing lens, and others with slow-growing optic cup and fast-
growing lens ; and either sort in both kinds of host. In every
case, there was a mutual influence of the components upon
each other. The growth of the eyes as a whole was of inter-
mediate rate, that of the rapid component was slowed down,
that of the slow component speeded up.2
Fig. 87, constructed from Harrison's data, shows the results
quantitatively. The association of a fast-growing optic vesicle
with a slow-growing lens retards the growth of the optic vesicle
by 22 per cent, in a punctatum body (No. 2 in Fig. 87 : ratio
1-26 instead of i-6i) ; and by 17 per cent, in a tigrinum body
(No. 4 : ratio 0-83 instead of i-o) ; similarly, it accelerates the
growth of the lens by 13 per cent, in a punctatum body (No. 9)
and by 6 per cent, in a tigrinum body (No. 11). The reverse
1 It is interesting to note that Twitty and Schwind (1. c.) in their
latest paper, published after this chapter was first written, also adopt
the phrase partition-coefficient of growth. They, however, think of the
partition-coefficient as a percentage ratio, and therefore, since the
ratio eye-diameter : body-length decreases considerably during larval
life, find it necessary to postulate a progressive change in the coeffi-
cient. If our conception be adopted of a partition-coefficient which
is an exponent of body-size, this difficulty disappears, and the coeffi-
cient can be considered to remain constant throughout growth.
2 That such interaction of neighbouring parts as regards their growth-
intensity need not always occur is shown by the recent work of Schwind
(193 1), who made heteroplastic grafts of parts of the shoulder-girdle
between the same two species. He found that the presence of a
grafted coracoid region growing at a different rate from the host's
organs had no effect on the growth of the host's scapular region. Thus
we have here an example of mosaic growth-rates, as opposed to mutual
regulation, which is perhaps comparable to the mosaic differentiation
of some developing eggs as opposed to the regulatory capacities of
others.
196
PROBLEMS OF RELATIVE GROWTH
16 ~Q) T whole eye on P
T whole eye on P Qy~
15
14
13
1-2
-(g) T optic vesicle onP
-(3) T lens ectoderm on P
II ■
T lens ectoderm onP($y
T optic vesicle on P (9)-
association, of a slow-growing optic vesicle with a fast-growing
lens, accelerates the growth of the optic vesicle in a tigrinum
body by 25 per cent. (No. 5), and by 22 per cent, in a punc-
tatum body (No. 3). And it retards the growth of the lens
by 20 per cent, in a punctatum body (No. 9), by 21 per cent,
in a tigrinum body (No. 10). The regulation is a gradual one,
as shown by Harrison's Fig. 37.
Further, Harrison gives conclusive proofs that the regula-
tion is due to chem-
ical and not in any
appreciable degree
to merely mechani-
cal causes. He also
finds some regula-
tion of a similar sort
in other tissues in
the proximity of the
eyes.
We could inter-
pret these results in
various ways. Per-
haps both rapid-
growing and slow-
growing organs pro-
duce specific sub-
stances which in-
teract with each
other. Perhaps
there is only one
type of substance, a
positive or growth-
promoting sub-
stance ; in such
case, when there is
a deficiency of this
in, e.g., the slow-growing lens, there will be a greater differ-
ence in concentration of this substance between optic vesicle
and lens, therefore more diffusion of the substance left in the
optic vesicle and accordingly less rapid growth of the opt c
vesicle. This latter suggestion would be in harmony not only
with the facts as to growth-centres and growth-gradients here
set forth, but also with recent work on plant growth, e.g. by
Dolk, who finds a specific growth-hormone with quantitative
10
0 9
0 8
0 7
0 6
@P lens ectoderm on T
(§)P optic vesicle on T
-(6) P whole eye on T
P opt/c vesicle on T (jdy^
P lens ectoderm on T(Th
P whole eye on T Qz)
(a)
(b)
Fig. 87. — Diagram to show the results of grafting
whole eyes and their parts between Ambly stoma
tigrinum (T) and punctatum (P).
The ordinates give the ratios of the linear dimensions of (a) the
optic bulb and (fc) the lens on the side receiving the graft to those
of the intact eye of the other side (see text for details).
(Constructed from Table II in Harrison, 1929B.)
EXTERNAL CONDITIONS 197
action. On this, however, only further experiment can decide ;
but the results are clearly of importance in that they show
experimentally that, as we have deduced from mere measure-
ments, rapidly-growing organs may exert an influence upon
the growth of neighbouring organs.
§ 7. The Influence of External Conditions
We next have to consider the effect upon growth-gradients
and heterogenic organs in general of the natural or experimental
alteration of conditions.
The heteroplastic experiments with Ambly stoma eyes which
we have just been discussing provide us with some facts bear-
ing on this problem. In Chapter II reference was made to
Twitty's experiments in grafting on to punctatum host-larvae,
tigrinum eyes of the same size but less physiological age than
the host's eyes.1 He found that when the hosts were starved,
their body-length (five animals, over an average of thirty-five
days) decreased by 18 per cent., their own eyes remained un-
changed in diameter, and the grafted eyes increased very
slightly, by 2-5 per cent. When, however, they were fed just
more than enough to maintain their size, their body-length
(four animals over an average of thirty-one days) increased
by 7 per cent., their own eyes by practically the same amount
(6-5 per cent.), but the grafted eyes by 40-5 per cent. One
of these specimens showed an increase of 50 per cent, in the
diameter of the grafted eye, but without any change in body-
length. It is thus clear that competition for food plays an
important part in the regulation of relative size of parts. The
rapidly growing eye, whether rapidly growing by virtue of its
youth or its higher specific growth-intensity, can draw dis-
proportionately upon the supply of food available (for further
discussion of this general problem, see Jackson, 1925, and
Huxley, 1921). It is clear that much valuable work would
be possible in determining the quantitative growth-relations,
in varying nutritive conditions, of heteroplastic grafts of the
same and of different ages, and of homoplastic grafts of
different age from their host. (Fig. 31.)
We have already considered the effect of poor nutritive
conditions upon the forceps of male Forficula (Chapter II) and
upon the dorsal crest of male newts (Chapter VI). Perkins
(1929) has analysed the effect of the parasite Sacculina upon
1 A similar regulation of rate of development was found by Choi
(193 1 ), who grafted lateral half -larvae of two species of frog together.
198
PROBLEMS OF RELATIVE GROWTH
the growth-gradient in Carcinus. He finds that the effect of
the parasite is primarily to reduce general growth of the crab
D
Fig. 88. — Regulation of growth of eyes of one species of Amblystoma grafted
to individuals of another species and of different age. All x 6.
G, grafted eyes.
(A) A specimen of A. punctatum with an eye from a younger specimen of the larger species, A.
tigrinum.
(B) The same 40 days later, after feeding on maintenance diet ; the grafted eye has enlarged.
(C) A similar specimen to (B), but starved since the operation ; the grafted eye has not grown.
(D) A. tigrinum, with an eye from an older A. punctatum.
(E) The same specimen 45 days later, after normal feeding. The grafted eye has lagged in its
growth.
EXTERNAL CONDITIONS 199
by acting as a new and a very active ' growth-centre ' which,
of course, enters into competition for food-material with the
organs of the crab's own body. Secondly, however, he finds
that this effect is not uniform on all the organs. It is greatest
in those of most rapid inherent growth-potential (highest
growth-coefficient). Thirdly, he believes that normally excess
of growth of one part is associated with a ' drainage effect '
upon the growth of other parts, and that when the growth
of the organs with high growth-coefficient is inhibited, as in
Sacculina, there is a releasing effect on those of low growth-
coefficient leading to an actual increase in their relative size.
It is by this means that he would account for the well-known
fact of the increase in the abdomen-width (approximation to
female type) in sacculinized male Crustacea.
This third conclusion of his does not appear to be well
founded. It is in disagreement with the correlation we have
found between the presence of a highly heterogonic organ
(e.g. chela) and excess growth in the appendages posterior to
it ; and with the important fact discovered by G. Smith
(1906A), to which sufficient attention has not been paid, that
parasitized female crabs, even if Perkins be right in supposing
that their general growth is checked, show accelerated develop-
ment of the female secondary sexual characters of their abdo-
men. This last point, together with the fact that in males
which have recovered from the parasite, the gonad develops
ova, seems to me conclusive proof that the effect of the parasite
here is in major part a specifically sexual one, and not merely
a quantitative growth-effect. Be this as it may, Perkins
has shown that there does exist such a quantitative growth-
effect of Sacculina upon its host, and that in regard even to
appendages which show no obvious secondary sexual differ-
ences, it exerts a differential and graded influence ; and this
opens the door to all kinds of future work.1
Another influence upon differential growth is shown by
the work of Thiel (1926) upon the Lamellibranch mollusc
Sphaerium. He shows that in clean water with high oxygen
content, the form of the shell becomes more rounded than
in the dirty water of its normal habitat. This, however,
1 In this connexion the specific and differential morphogenetic effect
of the parasitic worm Mermis upon its ant hosts should be noted,
although for the moment its relation to the growth-gradients of the
ant-body cannot be clearly envisaged. See Wheeler (1928), Vandel
(i93o).
200 PROBLEMS OF RELATIVE GROWTH
appears to be mainly or solely a mechanical effect due to an
alteration in the direction rather than the amount of growth
of the mantle-edge, and to be determined by an increase in the
number of embryos contained in the gills.
Similarly, Huntsman (1921) found that light not only
markedly retarded the growth of mussels (Mytilus), but also
affected their shape. Although the depth of the shell relative
to length was not altered, the relative breadth was increased
from 42 to 46 per cent ; as result, the ratio of weight to the
product of (/ X d X br) of course increased. The light
appeared to inhibit the growth of the edge of the mantle, which
is responsible for additions in length and in depth. Relative
breadth, on the other hand, again depends on the angle at
which the mantle-edge is growing.
Entz (1927), in an exhaustive study on Peridinians, has
shown that the relative length of the horns is markedly altered
by salinity, while the size of the body is little affected. In
different external conditions, the antapical horn may vary in
the ratio of 4 to 1 ; the post-equatorial horn in the ratio of
5 or 6 to 1 ; and the other two horns from zero to a con-
siderable development. As Entz puts it, the relative size of
the horns renders small differences of salinity visible in an
obvious and exaggerated form. (See also Gajewski, p. 259.)
What again appears to be a comparatively simple influence
is that revealed in what Przibram (1925) calls the ' Tail-
thermometer ' of rats and mice — namely, the fact that the
relative tail-length of these rodents (pp. 22, 40) increases
with increase of the external temperature in which they grow
up. The relationship is an approximately linear one. If rela-
tive tail-length be expressed as trunk length per cent, the value
of this index of tail-length (for young Albino rats) decreases
by 075 for each rise of i° C. in external temperature. He has
also shown that this is correlated with an actual increase of
internal temperature (0-2° C. for each 50 of external increase) ;
and further, that this is really the important agency in pro-
ducing the effect. We may therefore conclude that the effect is
due in part to a differential temperature-coefficient for tail-
growth. On the other hand, the fact that the tail (and other
organs such as ears and limbs, which also show a similar effect
with temperature) have a relatively large surface, and therefore
a lower temperature than the rest of the body (cf. Crew's and
Moore's proof of this fact for the mammalian scrotum), doubt-
less enters into the picture. Higher internal temperatures,
especially with higher external temperatures, would reduce the
EXTERNAL CONDITIONS 201
temperature-difference between tail and body, and would
therefore accelerate the tail's relative growth.1
Information perhaps more important, and certainly more
directly correlated with Perkins's results, is afforded by our
knowledge of the differential effect upon bodily proportions
produced by malnutrition, by hormones, and by teratological
embryonic agencies.
Jackson (1925) has published interesting data concerning
the effect of malnutrition upon differential growth in rodents.
An important fact is that in young animals held at constant
body-weight, certain organs, notably the skeleton, continue
growth, and continue on the whole in the same differential
way as in the normally-fed animal — a pretty demonstration of
the existence of inherent growth-potentials and their variation
in different parts of the same organ-system.
The actual details of the process vary according to the age
at which the animals are subjected to the treatment. (In
acute inanition leading to actual weight-loss, the changes, as
expected, are different in degree.)
In albino rats kept at maintenance from three weeks of
age, the tail actually grows in absolute length, and consider-
ably in relative length. The absolute head- weight is slightly
increased. When the experiment started at birth, the increase
in head-weight was much more marked. The brain and the
testes are other organs which show increase of weight when
growing animals are kept at constant weight, especially in
the new-born (Fig. 89). Full details are given by Jackson.
Hammond (1928, 1930, 1931) has also some interesting
remarks on the effect of undernutrition at various periods
upon relative growth in mammals (sheep, rabbit, etc.). In
an adult, the loss of weight occurs first mainly from fat, then
mainly from muscle, last of all the bones and the more ' vital '
organs, such as heart, lungs, etc.
In a young growing animal the vital organs appear to have
first call on nutritive material : during moderate under-
nutrition they thus continue their growth more or less un-
affected, while that of muscles (and fat) is markedly inhibited :
since muscles show positive heterogony in juvenile life, the
animal will retain juvenile proportions (and will, of course,
not be so valuable as a carcass).
In general, when organs have their period of maximum
absolute growth at different times (from whatever physio-
1 See also Abe (1931), Endokrinol., 9, and Przibram, ibid., no, for
effect of internal temperature upon relative cell-size in rats.
202
PROBLEMS OF RELATIVE GROWTH
logical cause), ample nutrition at the period of maximum
growth, followed or preceded by undernutrition at other
periods, will produce an animal with the organ of unusually-
large relative size ; e.g., in rabbits, the ears have an early
period of maximum growth : accordingly maximal feeding
when young, when followed by low feeding later, will produce
a rabbit with ears above the normal proportionate size.
Hammond further points out that since in meat animals the
maximum percentage of
flesh is only produced by
a high level of feeding
throughout most of the
growth-period, animals
cannot show their full
potentialities as meat-
producers if kept in un-
favourable nutritive con-
ditions. When practising
genetic selection for meat
qualities, it is therefore
necessary to provide
optimum conditions for
individual development
in order to be able to
pick out extreme vari-
ants. Conversely, if a
breed is required for a
poor environment, selec-
tion should be practised
with sub-optimal nutri-
tion, of the type to be
expected in this environ-
ment.
In regard to plants,
numerous examples are
known of external conditions affecting proportions. We may
cite two papers of Pearsall and Hanby (1925 and 1926). In the
former they showed that in Potamogeton leaves excess calcium
causes a predominance of growth in breadth, excess potassium a
predominance of growth in length. This was due partly to
the greater relative length of the individual cells in the Ca
cultures, partly to the arrangement of cells in the young
leaf-initials. In the latter paper they found that reduced
Head
£0.6 per cent
Head
££.T per cent.
Head
10. 1 per cenr.
fbre -limbs
6.9 per cenf.
Hind-limbs
15.6 percent
fbre-Iimbs
9.6 percent.
fore-limbs
8.5 per cent.
Hind-limbs
15.7 percent.
Hind limbs
15.4 per cent.
Trunk
6T.4 percent.
Trunk
54. 1 per cent".
Trunk.
55.4 per cent;
Controls at 3 weeks Constant > 10 weeks Controls at 10 weeks
Fig. 89. — Diagram showing changes pro-
duced in relative weight of various parts
of the body in young rats held at constant
weight for 7 weeks by under- feeding (centre)
as compared with controls at the beginning
(left) and end (right) of the experiment.
RHYTHMICAL IRREGULARITIES 203
hydrostatic pressure causes palmate leaves to become not
only smaller, but more dissected and with relatively smaller
basal lobes. However, as growth in plants clearly follows
very different laws from growth in animals, we will not
attempt to enter into any detail.
§ 8. Rhythmical Irregularities of Growth-ratio
Finally, we must just mention a curious but possibly im-
portant fact, namely the tendency of structures to grow by
abnormally high growth first in one dimension of space, then ,
in another, so that the growth-ratio for breadth on length does
not remain constant, but fluctuates more or less rhythmically
round a mean. Przibram (1902) has pointed this out in regard
to the carapace of crabs. In these he followed individuals
through several moults, so he could be certain of the fluctua-
tion in proportions.
I. W. Wilder (1924) finds a similar alternation, of periods
of filling out and periods of rapid length-growth with drop
in the body-build index, in the larval growth of the salamander
Eurycea bilineata, and the same phenomenon is stressed for
human children by various authors, notably Bean (1924),
Harris (1931 A and b), and with all the apparatus of biometrical
method by Berkson (1929) 1. Thiel (1926), as the result of
careful measurements on the skull of the bivalve mollusc
Sphaerium, finds an alternation of periods of increased intensity
of growth in height and decreased intensity of growth in
breadth with periods where the reverse relation holds. Cog-
hill (1928) finds evidences of the same sort of thing for definite
centres within the embryonic nervous system of amphibia.
(He also finds evidence that this periodicity alternates in
closely adjacent centres.) And various authors have empha-
sized that the same phenomenon occurs in regard to the post-
natal length- and breadth-growth of the human cranium.
Thus the phenomenon would appear to be general. This
again emphasizes the need for large bodies of observations
on proportionate growth ; but its physiological basis is at
present quite unexplained.
This chapter, as I pointed out in the introduction to it, has
inevitably been both discursive and inconclusive. However,
the facts and ideas set forth in it may serve to point the way
to more crucial and more radical methods of analysing the
problem.
1 See also" the recent paper of Davenport (1931), Proc. Amer. Philos.
Soc, 70, 381.
CHAPTER VII
BEARINGS OF THE STUDY OF RELATIVE
GROWTH ON OTHER BRANCHES OF
BIOLOGY
IT will be clear from preceding chapters that the study of
relative growth has important bearings upon many other
branches of biology. In the present chapter I will attempt
to summarize a few of these as succinctly as possible. We
will begin with its bearing upon systematics.
§ i. Heterogony and Taxonomy : Sub-species and
Taxonomic Forms
Systematists are agreed that mere size-differences may have
no taxonomic significance, since they are often the direct
effects of environmental conditions. But they usually attach
much greater importance to differences in the percentage size
of parts. Our studies of heterogony, however, make it obvious
that such differences may have precisely as little taxonomic
significance as those in absolute size, for wherever an organ
is heterogonic, differences in absolute total size of body will
bring about differences in relative size of the part.
When such problems crop up, they require analysis along
the following lines :
(i) Is there a difference in total absolute size between
the varieties considered ?
(2) If so, is the difference due (a) wholly to environmental
differences, (b) wholly to genetic differences, or (c) to a com-
bination of the two ?
(3) Is there a difference in the relative size of any parts or
organs ?
(4) If so, is there also a difference in total absolute size ?
(a) If not, heterogony is not at work, and the difference
is genetic and is of taxonomic value.
(b) If yes, then do the relations of relative size of part to
absolute size of whole obey the simple heterogony
law ?
204
HETEROGONY AND TAXONOMY 205
(i) If not, then some at least of the difference in
relative size (percentage proportions) is prima
facie of independent genetic origin, and there-
fore of taxonomic significance,
(ii) If yes, then prima facie the differences in pro-
portions are merely secondary effects of the
difference in absolute size, and therefore not
of taxonomic significance.1
(5) While measurement and simple mathematical analysis
will thus give important prima facie evidence, it is of course
desirable to apply experimental tests wherever possible to
clinch the matter.
A few examples will illustrate my meaning.
(1) The Red Deer (Cervus elaphus) of Scotland rarely exceed
125 kg. in (clean) body-weight, and rarely show more than
twelve points on their antlers. In some localities, the usual
run of body- weight is only 75-90 kg., with six to eight points.
In some districts of Europe, however (e.g. the Carpathians), stags
run to double the maximum Scottish body- weight, with twenty,
twenty-five or more points, and there appear to be no con-
tinental districts in which the mean weight or point-number
is as low as in Scotland. Further, the relative size (weight)
of the antlers is clearly much higher in the larger continental
stags (Huxley, 1931A).
It would be natural to consider the low size and antler
development of the Scottish deer as constituting criteria of
a true geographical variety or sub-species, as has also been
done with some island races, which are also small in both these
respects.
The problem is complicated, however, by the fact that in
the peat-bogs of Scotland are found the skeletons of deer
rivalling the biggest existing European specimens in size and
point-number ; and that these date back only a few thousand
years (Ritchie, 1920) ; Fig. 90. It would seem difficult to
imagine that true evolutionary (genetic) change reducing the
body-weight by half could have occurred in such a short space
of time. It is true that persistent killing for trophies of the
biggest stags, with the best heads, would have had a genetic
effect, but this could only have been intensively operative for a
very few centuries at most ; and in any case the same process
should have been at work on the Continent. Ritchie himself
1 See Champy (1929, p. 239) where the same problem is discussed
along somewhat similar lines.
206
PROBLEMS OF RELATIVE GROWTH
(1. c.) ascribes the change to the altered environment produced
by the cutting down of the forests in Scotland, and the giving
over of many of the original haunts of the deer to agriculture.
The Red Deer is by nature a forest species ; driven from the
lowlands, and when deprived of the shelter and succulent
browsing provided by forests, it grows stunted. This, how-
ever, takes no account of the difference in relative size of
antlers between Scotch and e.g. Carpathian strains. Here,
however, analysis of the available weight-measurements
Fig. 90. — Differences in absolute body- size and relative antler-size of pre-
historic (left) and modern (right) Scottish Red Deer.
(Huxley, I.e.) reveals, as we have seen (p. 42), that in spite
of large individual variations, the relative weight of antlers
increases regularly with absolute weight of body (Figs. 25, 29).
The antlers thus constitute a heterogonic organ, and we should
expect stunting to be followed by a diminution in relative antler-
size. Furthermore, the number of points on the antler is
also found to be a function of absolute antler-size (Huxley,
1926, 1931A). Here again there is considerable individual
variation, but the means show a regular curve. It would
appear to be an automatic consequence of increase of antler-
HETEROGONY AND TAXONOMY 207
bulk that the rapidly growing bone-material should branch
more profusely : the same phenomenon is found, e.g., in the
antler-like mandibles of stag-beetles (Lucanidae) and the
horns of other beetles (see Fig. 93).
We thus arrive at presumptive, prima facie evidence for
the truth of Ritchie's view. Finally comes experiment, and
settles the question. During the last hundred years, a number
of Red deer from Scotland (and elsewhere) have been imported
into New Zealand and there liberated. The regions where
they were liberated were profusely forest-covered ; and the
stags of Scottish strain in this new environment attained
body-weights of 200 kg. and over, with relatively large antlers
showing twenty and more points : in other words, they bridged
most of the differences between the Scottish and Carpathian
strains in one generation. Still further evidence is provided
by later events. Lacking natural enemies, the herds of deer
in New Zealand became too numerous, and began to destroy
the forests ; with overcrowding and less favourable food
conditions, ' degeneration ' set in, and now in many regions
the mean and the maximum body-weight and point-number
are far below what they were a few decades back. See Thomson
(1922) and Huxley (1931A).
We are thus perfectly justified in concluding (1) that the
bulk of the observable differences in size and proportions
between the existing Scottish strain of Red deer and (a) the
existing Carpathian strain and (b) the sub-fossil Scottish type
are non-genetic and of no taxonomic significance. (2) However,
we must admit that killing of the finest males is likely to have
had some genetic effect ; this will have been exerted on both
strains, but may have been more intensive in Scotland. In
any case, its primary effect will have been to reduce the mean
size of both races, the effect on relative antler-size and point-
number being consequential. It should further have had a
' dysgenic ' selective effect, in that among stags of equal body-
weight, more of those with genetic tendency to relatively
larger antlers will on the average have been killed off. (3)
There may also be true genetic differences between the Scottish
and Carpathian strains. But (4) the differences due to (2) and
(3) are relatively small, and the whole problem must be studied
afresh before we can be sure what they are, and indeed, as
regards those under (3), whether they exist at all. As corol-
lary, we may take it that differences in proportional antler-
size in dwarf island races of deer are probably only secondary
2o8 PROBLEMS OF RELATIVE GROWTH
effects of differences in absolute size, and that when an island
race is smaller, its small size is probably due in the main to
environmental stunting. This is supported by the findings of
Antonius (1920), who shows that in the dwarf race of deer on
Sardinia, the length of the face relative to the cranium is
reduced. Since the face is positively heterogenic, this again
is to be expected if the animal is stunted. The stunting has
produced a strain which, because dwarfed, has permanently
juvenile proportions. See also Klatt (1913) on dog skulls.1
I have spent some time on this case, as its analysis is fairly
complete. We may presume that similar differences in rela-
tive antler-size found in geographical races of Roe-deer are
secondary, though here the differences in absolute size
(which on the whole grades upwards from west to east across
Europe) are themselves more likely to be genetic, since the
type of habitat of the Roe-deer is approximately the same
over its whole range.
It is equally clear that many other percentage measurements
— e.g. of chela-size or abdomen-size in crabs — will in them-
selves be of no systematic value. What are taxonomically
distinctive in such organs are their growth-constants — the
values of the growth-ratio and of the partition-constant in
the heterogony formula, and the absolute body-sizes at which
heterogony (if it is not continuous or uniform) begins, ends
or alters. These numerical values, if properly established,
would be true taxonomic characters ; since, however, they
are far less convenient to ascertain (though probably of much
greater biological importance) than the usual type of specific
difference, which is selected for its obviousness and ease of
recognition, they are not likely to be much employed.
A peculiarly interesting example, constituting in a sense a
test-case, is provided by the Lucanid beetle Cyclommatus
tarandus, with markedly heterogonic male mandibles. This
has been studied by Dudich (1923), whose measurements and
conclusions have been further analysed by me (1927, 1931c).
Coleopterists have distinguished a number of forms of the male
of this stag-beetle — five main types, or if we include sub-types,
no fewer than seven, all based on structural characteristics of
the mandible. (See also Griffini, 1919, Natura, 10, 13.)
They are as follows (I omit the sub-types) :
1A slightly different case of retention of juvenile proportions is
given by Drennan (1932) who discusses the skull of the pre-Bushman
race of South Africa. This appears more akin to paedomorphosis (p. 239) .
HETEROGONY AND TAXONOMY
209
(1) Prionodont. Mandible beset with inwardly-directed teeth
throughout its length (2 sub-forms).
(2) Amphiodont. A toothless gap on the mandible, separat-
ing ' prebasal ' and ' sub-apical ' teeth.
(3) Telodont. Pre-
basal teeth lacking.
(4) Mesodont. With
a new, single 'sub-
median ' tooth, on the
ventral side (2 sub-
forms) .
(5) Mesamphiodont.
Numerous teeth in
homologous position
to sub-median tooth
of (4) ; and also a set
of teeth in position
of pre-basals of (2).
(Fig. 91.)
These types grade
into each other imper-
ceptibly, and the dis-
tinctions are admit-
tedly arbitrary : e.g.
a specimen is classified
as Amphiodont if a
gap is present which
is considered larger
than the normal gap
between two teeth,
Prionodont if it is con-
sidered not to exceed
this size. Further-
more, there is a
correlation between
tooth-type, mandible-
size, and body-size,
the five types, in the
order listed above, succeeding each other with increase of size '>
and even within each ' form ' there is a trend, with increasing
size, away from the type of the previous form and towards
that of the one succeeding.
It might be at once concluded, one would think, that all
14
Fig. 91. — (1) Female and (2 — 6) five different-
sized males of the stag-beetle, Cyclommatus
tarandus, to scale, to show the change in form
and relative size of the male mandible with
increase of absolute body-size.
Male forms : 2, prionodont ; 3, amphiodont ; 4, telodont ;
5, mesodont ; 6, mesamphiodont.
210 PROBLEMS OF RELATIVE GROWTH
the ' forms ' are purely arbitrary growth-types, the heterogonic
increase of mandible-size with total size being accompanied,
as often, by changes in tooth-characters and other morpho-
logical alterations. But Dudich states that he is forced to
regard them as in some way ' real ' and of taxonomic im-
portance, for the following reasons : (i) Because the several
' forms ' are transgressive as regards body-length, the smallest
of form 3, for instance, being smaller than the largest of
form 2. (2) Because, although the frequency-curve for female
body-length is unimodal and approximates to normal type,
that for the body-length of the males is irregular, multimodal,
and skew. He attaches more importance to (1).
Analysis (Huxley, 1931c) of his figures by body-size (178
specimens grouped into 28 classes) and a plotting of the
resultant means showed that when all the ' forms ' were
lumped together, a clear approximation to the simple hetero-
gony formula (constant growth-partition coefficient of male
mandible) was found (with the deviation from it at highest
absolute sizes which we previously established as usual in
holometabolous insects : see Chapter II). When however the
' forms ' were taken separately, and each subdivided into size-
classes, the resultant plot resembled a series of overlapping
tiles on a roof, thus giving graphic proof of Dudich's contention
that they are transgressive as regards body-size (Figs. 35, 92).
The type of overlapping, however, at once gives the key
to the situation. (1) What we have plotted are the means.
(2) But we should expect a definite range of variation in the
relative size of mandible for given body-size, whether this be
due to variations in growth-partition constant (k) or fractional
constant (b). (3) Over the range of size where two ' forms '
overlap (as elsewhere, of course) we should accordingly expect
to find some with mandible-size well above the mean for that
particular body-size, others with mandible-size well below
that mean. (4) But it is a general rule that increasing absolute
size of many organs (other Lucanid mandibles — see especially
the paper by Griffini, 1912 — deer antlers, etc.) is correlated
with morphological changes (Fig. 93). (5) We should there-
fore expect that, over each region of the size-range where
overlap occurs between two ' forms ', individuals which vary
in the direction of high mandible-size should on the whole
have the tooth-characters of the form above, those which are
low variants as regards mandible-size should on the whole
have the tooth-characters of the lower form. (6) The trans-
20 30 40 50 60 70 80 mm.
Fig. 92. — Graph to show relative growth of the mandible in the different male forn
of the stag-beetle, Cyclommatus tarandus.
Mean mandible-length (mm.) has been plotted logarithmically against mean body-length (mm.) for various si2
classes of each of the five male forms (see Fig. 91) : x , prionodont ; +, amphiodont ; Q, telodont ; A. mesodon
© tmes amphiodont.
The various curv« overlap each other in a regular way (see text).
211
212
PROBLEMS OF RELATIVE GROWTH
gressive variation of the forms as regards size has therefore
no ' real ' or taxonomic meaning, but is an automatic result
of splitting up a continuous series on the basis of purely arbi-
trary characteristics. (7) Thus, if you were to plot the extreme
variants, you would find the means for the series taken as a
whole keeping in the middle between the extremes, while the
curves for the class-means of the ' forms ' taken separately
would each begin
close to the ex-
treme plus limit
of variation, and
end near the ex-
treme minus limit.
I have treated
this case also in
some detail, since
it provides an
excellent example
of the dangers
inherent in mere
labelling : although
the several forms
are actually quite
continuous one
with the other as
regards their diag-
nostic characters,
the mere fact of
having separated
them, though on
arbitrary grounds,
made it possible
to group the facts so as to make the arbitrary separation
appear based upon some biological reality ; and it is only
further analysis based upon a study of the laws of relative
growth and its variation which enables us to detect the fallacy.
§ 2. Heterogony in Groups higher than the Species
There is no need to imagine that the heterogony mechanisms
and other laws of differential growth are confined within the
boundaries of single species. On the contrary, there is every
reason to suppose that, like the activities of ductless glands,
these growth-mechanisms may operate in very similar fashion
Fig. 93. — To illustrate change of form of man-
dible with increase of its absolute size in male
stag-beetles.
Below, a, b, c, side view of the head in the minor, media and major
forms of Psalidoremus inclinatus. Above, a — 1, mandibles in the
genus Odontolabis.
HETEROGONY AND TAXONOMY 213
throughout considerable groups of animals. What we have
just been saying about systematics has an immediate bearing
upon certain evolutionary problems. For we should then
expect to find heterogonic organs varying in their relative
size with absolute size of body in different-sized species of the
same genus, or different-sized genera of the same family, just
as in different-sized individuals of the single species. Our
supposition is confirmed : this does occur. It is not universal
or inevitable, but it does occur frequently.
The rule was independently discovered by Lameere (1904 ;
seealso 1915) and Geoff rey Smith (1906B), and numerous further
instances of it have been put on record by Champy (1924
Fig. 94. — Phenomenon of Lameere and Geoffrey Smith in species of the
Dynastid beetle Golofa.
Below, two specimens of G. porteri.
Above, right, G. cczcum ; left, G. imperialis.
and 1929) : Lucanidae (1924, pp. 142, 152) ; Dynastidae (ibid.,
pp. 152-4) ; and see Fig. 94.
In the most striking cases, the heterogonic organs in the
males of the large species are relatively enormous, while in
those of the small species they are scarcely more developed
than in females (e.g. Fig. 94 ; the beetle Golofa porteri against
G. imperialis, Champy, 1929, p. 230). This at first sight appears
as an example of orthogenesis ; but if orthogenesis be taken
in its strict sense, of a determinate evolutionary change in
the germ-plasm, this is not so. All that we have is a type
of growth-mechanism, which, if not modified, will yield certain
predictable results, as regards relative organ-size, with any
given absolute body-size. If a new high or low extreme of
body-size, not previously attained in the evolution of the
214 PROBLEMS OF RELATIVE GROWTH
group, is reached by a given species, the growth-mechanism
will see to it that new potentialities of relative organ-size are
realized. But though the proportions of the organ are new,
the same single growth-mechanism is at work. There has
been no change in its hereditary basis, determinate or other-
wise ; merely new results, potential in it from the first, have
been realized.
If we wish to search for evolutionary parallels, we find that
a similar situation has occurred in regard to many time-
relations in body-processes of warm-blooded vertebrates,
which take place at temperatures well above the lethal tempera-
tures for most cold-blooded vertebrates. Accordingly such
processes occur at speeds which are never realized in cold-
blooded forms. The speed of such processes in the cold-blooded
forms varies with temperature in a regular way. The maxi-
mum speed in the cold-blooded forms is limited by the tempera-
ture which they can tolerate without dying ; but the actual,
and much higher, speeds in warm-blooded forms approximate
closely to the speeds which are obtained by extrapolating the
curve obtained for the cold-blooded forms to the blood-
temperature of the warm-blooded. An example of this is
seen in the lapse of time between onset of convulsions due to
insulin injection in a frog, which dies at about 30 ° C, and
in a rabbit, which dies at 370 C. (Huxley and Fulton, 1924).
The evolutionary importance of the facts lies in this : that
whenever we find the rule of Lameere and Smith holding true
for a series of separate forms, we are justified in concluding
that the relative size of horn, mandible, or other heterogonic
organ is automatically determined as a secondary result of a
single common growth-mechanism, and therefore is not of
adaptive significance. This provides us with a large new list
of non-adaptive specific and generic characters.
The operation of the rule, however, is not constant : it is,
in fact, merely a rule, with numerous exceptions. There are
a number of cases in which related species of very dissimilar
absolute size show secondary sexual characters of the same
relative size. Champy (1924, p. 155) lists a number of these,
and Dr. Arrow, of the British Museum (Natural History),
has informed me that there are numerous other examples to
be found in Coleoptera. This should, however, not surprise
us. The fact that the growth-ratio of the male chela of Uca
changes during the life-history, or that the onset and end of
heterogony may vary for the same organ in different types,
HETEROGONY AND TAXONOMY 215
indicates that such growth-mechanisms can be quite markedly
modified should biological need arise.
We meet with a precisely similar state of affairs in regard
to other biological rules. For instance, there is an undoubted
tendency for secondary sexual characters acquired by one sex
to be transferred, in whole or in part, to the other sex, even
if of no biological value in the other sex. But this general
tendency is constantly being modified or overruled by other
agencies, e.g. the need for protective coloration in certain
female birds frequently inhibits the transference of male
display characters to the females, and the tendency can only
manifest itself, among females of these groups, in those species
which are not in need of protection as regards colour, because
they nest in holes (e.g. tits, red-breast).
Doubtless, as well as such total exceptions to Lameere's
rule, we should also find many minor modifications, revealed
by organs whose relative size was not precisely what should
be expected on an embracing formula, though in part a function
of absolute body-size. But to detect these we should have to
make quantitative studies of the variation of relative organ-
size (a) within the individuals of the several species, (b) among
the species of the genus ; and this has not yet been attempted.
The undertaking of this task in insects and Crustacea should
be all the more interesting, since in regard to relative brain-
size in mammals, Lapicque (1907, 1922) and Dubois (1914,
1922) and Klatt (1921) have established the remarkable fact
that different formulae apply to the infra-specific and inter-
specific variation respectively. Variation of body-size within
the species (e.g. in different breeds of dogs or rabbits) has
less effect on absolute size (though more on relative brain-size)
than does variation of body-size from one species to another.
In both cases the formula y = bxk is followed, but whereas
the value of the growth-partition coefficient (k) of the brain
is 0-56 when different species are considered, it is only
0-22 for different-sized individuals of the same species. In
this particular case, the facts are doubtless to be correlated
with the artificial nature of the intra-specific selection that
has been at work. Man has been concerned to produce large
or small breeds, irrespective of their general adaptation or
their efficiency in a state of nature. The changes in absolute
brain-size produced by such changes in body-size are presum-
ably the minimum to be expected as automatic or secondary
result of the change in body-size ; while those found as
216 PROBLEMS OF RELATIVE GROWTH
between wild species would represent the best possible physio-
logical adjustment between brain-size and body-size. We
should also expect that the genetic variations in body-size
within a wild species would also presumably be largely inde-
pendent of those affecting absolute brain-size ; but so far as I
know, data on this point are lacking. (See also p. 224 ; heart.)
Whether any such discrepancy between intra- and inter-
specific formulae would be found in regard, e.g., to the horns
or mandibles of male beetles remains to be seen. If it did
occur, it might well be expected to be of opposite nature, the
larger species having relatively smaller horns than would be
expected from the extrapolation of the intra-specific curve of
smaller species. But this, as I have already said, also remains
to be determined.
§ 3. Heterogony and Evolution
A further evolutionary implication of these facts is this :
that the existence of an organ with high growth-ratio tends
to limit the extreme size attainable by the type in question
during evolution, for with very large absolute size, the
relative size of the organ will tend to become so huge that
it becomes unwieldy or even deleterious. Doubtless the organ
could be held in check by a modification of its growth-ratio
such as we have been envisaging. But for increase of absolute
size, two concomitant processes of mutation would then have
to occur — mutations favouring decreased growth-ratio of the
organ, as well as those favouring increased absolute body-size,
so that the evolutionary problem is thereby complicated.
It can be calculated that a male Uca with body weighing
1 kg. would, if its claw's growth-ratio had remained unaltered,
boast a large chela weighing some 10 kg. — which, as Euclid
says, is absurd : and it is perhaps no coincidence that the
largest fiddler-crabs attain sizes far below those of many
other Brachyura, and even far lower than those of other land
or semi-land crabs (Ocypoda, Birgus, etc.) ; see p. 32. The
excessive antlers of the Irish ' elk ', Megaceros, may very
likely be accounted for on similar grounds. An organ which
is verging on the deleterious may rapidly become actually
deleterious if conditions change.
A further important modification of the rule has been
pointed out by Champy (1924, p. 156 seq.), viz. that it applies
only to the size of the organ. The details of form may be
radically modified, and yet the general size conform to the rule.
HETEROGONY AND EVOLUTION
217
This implies that we have a fundamental growth-mechanism
controlling size and location of organ, but that its form is
capable of modification by numerous independent genes. We
Fig. 95. — Specific variations in detail of heterogonic organ (cephalic horn)
in different species of Goliath beetles (Goliathidae).
1 Chelorina polyphemus. 2, Eudicella euthalia. 3, Goliathus giganteus. 4, Mccynorhina torquata.
5, Taurina nireus. 6, Stephanorhina guttata. 7, Neptunides polychrous. 8, Taurma Umgiceps.
q Trigonophorus delesserti. 10, Rhanzania bertolinii. 11, Myctenstes rhinophyllus. 12, Theodosia
magnifica. (1 1 and 1 2 resemble Dynastids in possessing prothoracic horns) . 1 3 and 14, Dicranocephalus
wallichii, two views.
must further suppose that other independent genes are capable
of modifying the size-relations (growth-ratio) of the organ in
minor quantitative ways. (Fig. 95.)
2i8 PROBLEMS OF RELATIVE GROWTH
Perhaps the most interesting evolutionary aspect of the study
of relative growth is when we find a series of related forms pos-
sessing a markedly heterogonic organ, which not only differ in
size, but also succeed each other in order of absolute size during
evolutionary time. A classical example is afforded by the
deer (Cervidae). It was long ago pointed out that in a broad
way the antlers of deer increased in complexity from their first
geological appearance up to the Pleistocene, much as do the
antlers of an individual deer during its single lifetime, and
this was adduced as an example of the laws of recapitulation.
So in a sense it is, but it appears to owe its existence to the
presence of a fundamental heterogony-mechanism for antler-
growth, accompanied by increase of body-size both during
ontogeny and phylogeny. Increase of absolute body-size
automatically brings about disproportionate increase of antler-
size, and increase of antler-size appears to be of necessity
accompanied by greater complexity of branching (or palmation).
And this applies equally to the individual and to the race.
Theoretically, the most interesting case is that of the
Titanotheres, worked out by H. F. Osborn (1929, and earlier
papers there cited). These, like so many other mammals,
begin their evolutionary career as small organisms, and steadily
increase in size until they become extinct. They also begin
hornless, and end with a single bifurcated frontal horn.
But — and this is the salient feature of this example — Osborn
has been able to distinguish at least four distinct lines of
descent in the group, characterized by differences in skull-
shape, dentition, leg-size, etc., and presumably mode of life ;
and in each of the groups we meet with the same phenomenon
of small and hornless forms steadily increasing in size and
eventually becoming horned. Thus we find the origin of
horns of the same type, growing from the same location, taking
place independently in four separate groups : and Osborn
insists that this must be interpreted as true orthogenesis — in
the strict sense of the term, as implying predetermined varia-
tion of the germ-plasm, not merely of directional evolution.
However, as pointed out earlier (Huxley, 1924; and see
Sturtevant, 1924), our studies make it clear that this interpre-
tation is not necessary. Granted (a) that there existed in
the germ-plasm of the ancestor of the four lines of descent
the hereditary basis of growth-mechanism for a frontal horn,
and (b) that increase of size up to a certain limit was advan-
tageous for Titanotheres in general, as would seem inherently
HETEROGONY AND EVOLUTION 219
probable, then the results follow without any need for invok-
ing orthogenesis. Natural selection would account for the
increase of absolute size, and increase of absolute size would
evoke the latent potentialities of the horn's growth-mechanism
in the history of the race just as surely as it does, for instance,
in the individual deer to-day, or as it undoubtedly did in the
individual Titanothere in the Oligocene. No alteration in the
hereditary basis for horn-growth need be involved at all,
whether determinate or otherwise : the visible horn-changes
are automatic results of the effect of size-changes in the un-
altered mechanism for horn-growth.1
The only remarkable thing about this case is that the
mechanism for horn-growth must have been represented in
the germ-plasm before it could ever operate — before any single
Titanothere was large enough to have a horn at all. And
this is undoubtedly surprising. Once more, it makes it ex-
tremely difficult to assign any adaptive value to the horns,
at any rate in the early stages of their evolutionary develop-
ment, when they were represented only by the merest incipient
knobs, and still more when they merely existed potentially ! 2
To take a case nearer home, Parsons (1927) maintains that
the British race is showing evolutionary change, because the
mean proportions of British skulls have slightly changed
1 Doubtless some quantitative modifications in growth-ratio of horn
took place, as evidenced by the difference in relative horn-size in the
various groups ; but this does not affect the main argument.
Champy (1924, p. 155) takes an almost identical view. He writes :
' Le phenomene d'orthogenese ou rectigradation, lorsqu'il accompagne
une augmentation de taille, n'est pas un phenomene evolutif, c'est
un phenomene purement physiologique d'equilibre nutritif. La seule
evolution qu'il y ait a expliquer est l'augmentation de taille.'
2 The Asiatic Titanothere genus Embolotherium is distinguished not
only by possessing a very large and peculiarly-shaped horn, but by
this being formed entirely from the nasal bones, instead of from both
nasals and frontals, as in all other genera. During both the ontogeny
and the phylogeny of other late Titanotheres, there is a tendency for
the horns to grow in an anterior direction as well as to increase in size.
It may be suggested that the condition in Embolotherium represents
a further step in this evolution — the actual shifting of the growth-
centre from the fronto-nasal to the nasal region. This would be a
shift of a whole subsidiary growth-region with a gradient of its own,
along a main gradient. Somewhat similar shifts of quantitative growth-
centres within a single growth-gradient have been already recorded.
The most relevant case is the shift of the growth-centre of the female
abdomen of Pinnotheres to the terminal segment in place of the sub-
terminal segment where it is situated in most Brachyura (p. 95) ; cf.
also p. 117 (Copepod body-regions).
220 PROBLEMS OF RELATIVE GROWTH
during historic times. This may be so ; but caution is indi-
cated. If it should prove to be the case, as is quite possible
on the evidence, that the various dimensions of the skull alter
at slightly different rates with increased absolute size, then
the change observed by Parsons may be merely the effect of
the increase of mean stature which appears undoubtedly to
have occurred in Europe in the past few centuries. And
since this is in all probability phenotypic in origin, the change
in skull-proportions, though of course in a sense an evolutionary
change, has far less significance than we should at first sight
be inclined to assign to it. (See Hooton, 1931.)
Parsons himself states that the mean male stature of the
upper and middle classes has risen rapidly by perhaps three
inches to about 5 ft. 9 in., whereas the mean for the whole
population is only about 5 ft. 5 in., and that for the stunted
slum and factory populations perhaps an inch or so less. The
high stature of the best-nourished classes is a modern pheno-
menon, due to good food, exercise, etc., and was not realized
among the early Saxons (mean 5 ft. 6 in.).
The change in skull-proportions to which Parsons refers is
one in increased relative height (taken as percentage-ratio of
(auricular) height in relation to total head-size — measured as
[length + breadth -f- height] as a standard). As a matter of
fact, when we make a correlation-table between relative
skull-height and absolute skull-size as measured by Parsons's
figure for [length + breadth + height], we obtain a definite
correlation (Table XIII).
Naturally it would be much more satisfactory to make a
correlation table for all the individual measurements available,
but this must be a task for the physical anthropologists.
We should also expect to find a correlation between total
stature and total head-size : it is noteworthy that the lowest
values are for poor neighbourhoods in London and for soldiers
in the eighteenth century ; and the educated twentieth-
century type, which we know to have a mean stature much
above that of previous centuries, has values both for absolute
skull-size and relative skull-height far beyond any previous
limit.
As regards relative breadth (cephalic index), it seems probable
that different skull-types may show a change with age (size)
in either of two opposite directions, but here the evidence is
conflicting. In any case Parsons's data are most simply,
though not necessarily, to be interpreted as a further ex-
HETEROGONY AND EVOLUTION
TABLE XIII
221
G
F
B, E
A
C
D
Relative skull-
height, per cent
272-6-277-5
267-6-272-5
262-6-267-5
257-6-262-5
252-6-257-5
460-1 465-1 470"1 475"1 480-1
-465 -470 -475 -480 -485
Absolute skull-size (1. + br. + h.), mm.
A Mean of the three main racial stocks of Britain, early Mediterranean
(Neolithic), Alpine (Beaker), and Nordic (Anglo-Saxons).
B Fourteenth and fifteenth century villagers, Northants.
C London plague skulls, seventeenth century, probably poor stock.
D Eighteenth century, London, poor neighbourhood.
E Soldiers, eighteenth century.
F Twentieth century, poor classes.
G Twentieth century, educated and well-to-do classes.
ample of change of relative proportions with change of absolute
size.
As suggested by Hooton (1931, p. 408), a similar effect may-
be responsible for the changes in skull-form of the descendents
of immigrants into the U.S.A., made familiar by the work
of Boas.
All the cases we have so far considered have another aspect
of interest. They are all examples of what Darwin called
correlated variation, where change in one character auto-
matically brings about change in another character (unless
counter-selection operates against the correlated change).
Thus they relieve us of the necessity for seeking utilitarian
explanations for the correlated changes ; natural selection
need not be invoked to account for the details of such char-
acters.
The burden on natural selection is also appreciably light-
222 PROBLEMS OF RELATIVE GROWTH
ened by the demonstration of the existence of growth-gradients.
As pointed out in Chapter IV, the method of Cartesian trans-
formation of related forms employed by D'Arcy Thompson
shows that the great majority of the striking changes in shape
and relative size of parts necessary for the evolution of a sunfish
(Orthagoriscus) from a Diodon-like ancestor not only need
not but cannot have been separately evolved ; they can only
be interpreted as due primarily to a single main change in
the form of the growth-gradient along the animal's axis. And
the factual evidence we possess, as well as a priori reasoning,
indicates that a mutation can act on a growth-gradient as a
whole, thus simultaneously altering the proportions of a large
number of parts.
An even more fully worked-out example is that of the con-
formation of domestic breeds of sheep analysed by Hammond
(I.e.). As we have already seen in Chapter III, Hammond
has been able to show that the changes in proportions which
take place in foetal and post-natal life in improved breeds of
' domestic ' sheep represent merely an accentuation of quali-
tatively similar changes which occur in unimproved breeds
and in wild species (Fig. 96). In other words, there is a
greater intensity of the relevant growth-changes, which means
primarily a steepening of the growth-gradients concerned.
Here again, the characters actually affected by selection are
quite few, namely the growth-gradients ; whereas the number
of what are usually called ' characters ' which are altered
during the process is very large.
In general, we may say that the existence of growth-gradients
gives opportunity for mutation and selection to affect a
number of parts in a correlated way, thus greatly simplifying
the picture of the genetic and selective processes at work.
The cases previously considered concern organs with specific
heterogony mechanisms. D'Arcy Thompson (1. c.) has drawn
attention at some length to the functional changes necessitated
by increase in absolute size. These are for the most part
well known, such as the need for increased relative weight of
skeleton in large land animals, the need for increase of absorp-
tive surface of the gut with increase of bulk to be nourished
(on which topic a recent essay of R. Hesse (1927) may be
consulted for further details), and so forth. In any case,
they do not so much concern us here, since we are chiefly
dealing with the automatic and primarily non-adaptive effects
of inherent growth-mechanisms.
HETEROGONY AND EVOLUTION
223
An exhaustive botanical study of the relation of size to
internal morphology has recently been published by Wardlaw
(1924-8), but the details are of a technical nature and I cannot
enter into them here. The conclusion which chiefly interests
us is that certain characteristic types of internal morphology
in vascular plants (e.g. polystely) need have no phyletic value,
since they appear to be modifications directly dependent upon
increase in size. These and other -'results have been sum-
Fig. 96. — Changes in proportions during growth in wild sheep and improved
breeds of domestic sheep (see text).
marized in a book by Bower (1930). The general conclusion
to be drawn from these botanical studies would seem to be
that in this case apparent orthogenesis is to be explained by
parallel adaptive evolution due to size-increase.
We may take one example of the individual changes corre-
lated with size increase (Bower, I.e., p. 16). As we pass up
from rhizome to stem in Psilotum triquetrum, the size of the
organ increases. The size of the tracheidal tract also increases,
and its form also changes, exposing more surface than it would
224
PROBLEMS OF RELATIVE GROWTH
have if it remained approximately cylindrical. This is summar-
ized in the following table, abridged from Bower's Table I.
Diameter of stele, mm.
Surface-volume ratio of tracheidal
tract
Surface-volume ratio of equiva-
lent cylinder
0-17
o-34
0-29
o\53
o-6i
9-26
5-90
5-40
3-64
3-20
6-82
4-!5
1-88
1-08
0-91
075
3-00
o-8i
i.e. the actual surface : volume ratio diminishes to about
30 per cent of its initial value : but if the conducting tissue had
remained cylindrical in form the ratio would have decreased
to about 13 per cent. Changes precisely similar in principle
occur in phylogenetic series with increase of size : see e.g.
Bower's Table X.
§ 4. Heterogony and Comparative Physiology
Disproportionate change of relative size with change of
absolute size may have important results in studies in com-
parative physiology. (See also Murr, p. 260.)
This is well brought out by Klatt in an interesting paper
(1919). He finds the relation of heart-weight to body- weight
in vertebrates to follow our familiar formula for constant
growth-partition, the (mean) coefficient, or as he calls it, the
' heart-exponent ', being 0-83 : i.e. heart-weight = constant
X body- weight °'83. It is interesting to find that the value
appears to be very similar whether the size-differences con-
sidered are intra- or inter-specific ; this result differs from
that of Dubois, etc. (1. c), on the brain, a difference which may
presumably be correlated with the prompt functional regula-
tion of the size of the heart to the work it is called upon
to do.
In passing, it may be noted that the heart as a whole appears
to have a different growth-coefficient from the right ventricle
and the auricles. These latter, according to Hasebrock (1927),
have a coefficient close to 2/3, which would indicate their
functional dependence upon surface-area, or upon general
metabolism, which in warm-blooded forms is nearly propor-
tional to surface-area ; while the left ventricle has a much
higher coefficient, nearly in direct proportion to body-weight.
But from the point of view of comparative physiology,
HETEROGONY AND PHYSIOLOGY
225
the interest lies in the following considerations. It has usually
been customary, as was done by Parrot (1894) for birds, to
compare the cardiac efficiency of different species according
to their percentage heart-weights. From what has been said,
this is entirely erroneous ; variations in cardiac efficiency
can only be determined in relation to the mean value to be
expected at a given body-size for the exponential growth-
partition formula, which will be given by the fractional co-
efficient b in the formula y = bxk. Clearly, the percentage
method will put large birds at a disadvantage.
TABLE XIV
True relative
A
Per cent.
B
heart-size =
heart-weight
fractional
constant (6)
Capercaillie
7-81
Magpie ....
0-0228
Magpie ....
9-34
Chaffinch
0-0240
Wild Swan .
1 1 78
Capercaillie
0-0322
Chaffinch
14-16
Hobby ....
0-0402
Peregrine Falcon .
14-91
Peregrine Falcon .
0-0467
Hobby . . .
16-98
Song Thrush .
0-0528
Song Thrush
25-64
Wild Swan
0-0550
The table shows for a few birds from Parrot's list the order
of cardiac efficiency (in ascending relative heart-size) as deter-
mined (A) by percentage methods and (B) by the correct
method. It will be seen how different the two are.
Dubois (1. c.) and Lapicque (1. c.) have given us a similar
method for judging of the true degree of ' cephalization '
(true relative brain-size) in vertebrates. Thus the true relative
brain-size (fractional constant b in the growth-partition formula)
for the mouse family is 0-07, for the cat family 0-31-0-34, for
tailed monkeys 0-4-0-5, for anthropoid apes 0-7, for man 2-8,
whereas percentage values give no information of any value
at all.
A similar method of approach can be utilized for many organs.
Huxley (1927c) has used it for relative egg-weight in birds.
Here it is found that the exponent (partition-coefficient of
material between egg and body) is not constant over the size-
range covered by birds, but diminishes steadily with absolute
size, from 0-9 or 0-95 for the smallest birds to about 0-7 for
15
226
PROBLEMS OF RELATIVE GROWTH
the largest. It is suggested that it would be most advantageous
to have a linear relation (exponent = i-o), but that physio-
logical difficulties connected with surface-area (which would
give a limiting value of 0-67) interfere with this, and that they
interfere increasingly with increased absolute size. It is inter-
esting to find the large Ratites lie very close to the theoretical
curve, in spite of their relatively
huge bulk and their flightless
habits. (Fig. 97 ; see also p. 264.)
Body-weight, q.
300 400 600 BOO 1.000
SO
Body-weight, q.
8 iol"PP«"»""20
1 ■
1 1 r
4 000 6000 8.000 10.000 70000 30.000 40,000 100.000
Body-weight q
(lower cur.el
Fig. 97. — Graph showing change of egg- weight with body-weight in 432 species
of birds : Means by weight-classes, logarithmic plotting.
o, Carinatae ; X, Ratitae (excluding Apteryx, which is plotted separately) ; K mean for last class
of Carinatae combined with Ratitae (except Apteryx).
The graph begins with the upper curve ; k = i-o. The value of k then sinks (dotted line), finally
approximating to 0-67 (solid line, below).
Each large group has its own curve, differing chiefly in the
fractional constant b. The modifiability of relative egg-size
in relation to adaptive needs is clearly seen in the abnormally
low position of the value for species which are reproductive
HETEROGONY AND PHYSIOLOGY 227
parasites upon smaller birds, like Cuculus canorus, and in
the high values of b found for most forms with precocial
young.
In regard to the physiological side of the problem, it is im-
portant to distinguish two very different ways in which change
in relative size of a part may be brought about. We have
already discussed certain implications of this distinction, but
have not set it forth in precise form. (See also pp. 257-8.)
(1) An organ may be capable of functional adjustment in
size. The best example of this is the heart.
(2) An organ may possess a differential coefficient of growth-
partition, which operates wholly or largely irrespective of
functional demands, e.g. the antlers of deer, the large chela
of male Uca, the limb of Ambly stoma tigrinum grafted on to
A. panctatum, the pituitary of rabbits (Robb, 1929), etc.
Increase of absolute size of an animal will in both cases bring
about an alteration of relative size of the part, but in the
first case owing to functional hypertrophy, in the second owing
to the specific growth-intensity of the organ, which in its turn
is presumably due to a specific growth-promoting substance.
And doubtless in many cases, both mechanisms will operate
simultaneously.
Some cases of so-called functional hypertrophy may turn
out to belong to the second group, e.g. the enlargement of the
remaining testis after unilateral castration is probably not
functional in the usual sense at all, but depends upon the exist-
ence of a specific partition-coefficient for testis-tissue relative
to total size. (See Domm and Juhn, 1927, and also p. 257.)
The distinction, as I have suggested, is usually not an
absolute one. If one prevents a limb from exerting its normal
function, e.g. by tying it up at birth, its growth will be markedly
subnormal. With regard to heart-weight, etc., it is a priori
almost certain that the heart, in addition to its capacities
for marked functional size and regulation, possesses a primary
partition-coefficient which will differ from species to species.
What we can say, however, is that in some cases the differences
observed are due solely or mainly to functional regulation,
in others solely or mainly to differences in inherent growth-
capacity.
But there are cases in which organs attain their definitive
relative size entirely owing to one or other of the two methods.
The size of the organs of holometabolous insects must depend
entirely on their growth-partition coefficient ; while it is
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RELATIVE GROWTH AND GENETICS 229
highly probable that the size of tendons in the mammalian
body is entirely or almost entirely due to functional response.
Thus, when we compare organs of different relative size in
two individuals, races, or species of different absolute size,
our analysis should be devoted to establishing (1) whether
the difference is due to the organ's capacity for functional size-
regulation ; (2) whether it is a mere consequential effect of
increase of absolute size, as is the case with the relatively
large antlers of Red Deer imported into New Zealand from
Scotland, or of the relatively large mandibles of absolutely
large male Lucanidae ; or (3) whether it is due also to genetic
alteration in the growth-partition coefficient ; or (4), of course,
to a combination of these in differing degrees.
§ 5. Relative Growth and Genetics
We next come to the subject of genetics. It is clear that
the chief genetic factors controlling any organs which show a
constant differential growth-ratio or a constant growth-
partition coefficient must be rate-genes, or genes which deter-
mine the rate of a developmental process. Such factors have
been deduced or demonstrated for numerous characters,
notably in the sex-determining genes of moths (Goldschmidt,
1923, 1927) and the eye-colour genes of the crustacean
Gammarus (Ford and Huxley, 1929) (for other cases see the
references in the works cited, and also Ford, 1929, Belehradek
and Huxley, 1930) ; see Figs. 98, 99.
There are two general points which are worth stressing.
The only case where the curve for the development of such
characters has been quantitatively obtained by direct measure-
ment is that of the eye-colour of Gammarus. Here it appears
clear that the actual form of the curve is determined by two
factors — an inherent greater or less velocity of pigment-
deposition, and some regulating mechanism which relates
this to the growth of the body as a whole. This is clearly
brought out by two facts — first, the relations in slow-growing
mutants ; secondly, the slight lightening of colour which
occurs in the eyes of certain strains of Gammarus after sexual
maturity. ' Slow growth ' is a recessive mutation, which at
first shows abnormally dark eyes. This appears to be due to
the fact that the eggs are of normal size, that a pigment
precursor is formed in them in some simple relation to egg-
size, and then, when the time comes for the deposition of
pigment, the eye-area, over which it is to be spread, is abnor-
230
PROBLEMS OF RELATIVE GROWTH
10-
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V
p
^ 2
-
-
-
-
_rrSS-
A—n
nss
-
/ /
5exu.il maturity
of dd begins
Extrusion 10
20 30
Time in Days ,
VO
50
Fig. 99. — Actual rate of darkening (smoothed curves) in 1,000 rapid-darkening
(n-SS) and 1,000 slow-darkening (rrss) specimens of Gammarus chevreuxi at
230 C. (compare Fig. 98).
mally small owing to the slow growth. Later, however, the
eye-colour is regulated towards what is ' normal ' for the
particular rate-genes involved; this appears due to the fact
that further formation of pigment occurs in relation to body-
size (Ford, 1928) ; Fig. ioo.
12
%>
<^ 5
I"
v
*
Time in Days
Fig. 100. — Effect of size (rate of body-growth) on rate of eye-darkening in a
single genetic strain of Gammarus chevreuxi.
All specimens were males of constitution rrss, kept at 230 C. The figures on the curves give eye-
length in mm.
Max., curve of darkening in slow-growing specimens, which attain sexual maturity at time repre-
sented by vertical line C ; min., curve for fast-growing specimens (maturity attained at A) ; mean,
curve for mean of population (maturity attained at B).
-
-
max..
mean.
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RELATIVE GROWTH AND GENETICS 231
The second point, the slight lightening of eye-colour after
maturity, is due to the expansion in size of the individual
eye-facets, causing the same amount of pigment to be spread
over a greater area (Ford and Huxley, 1927).
This complex relationship is precisely similar to what we
have found to obtain for the relative growth of an organ
such as a limb (see Chapter VI), where two factors are also
at work, the inherent growth-potential of the organ, and some
mechanism regulating this in relation to the growth of the
body as a whole. Since the relative factor occurs in both
cases, it is not to be expected that the curves for characters
controlled by rate-genes will shed light directly on the nature
of the biochemical processes involved, even if plotted against
time. It would be theoretically better to plot them against
some quantitative character of the animal, e.g. the amount
of some important substance in the body, or even the total
body-bulk. In this way we should obtain a growth-partition
coefficient for the chemical substance responsible for the
character, just as we obtain a similar coefficient for the growth
of heterogonic organs when their size is plotted, not against
time, but against body-size ; and it is these coefficients which
are of the greater biological importance, and are quantitatively
comparable one with another. That this method may be
valuable is seen in the already cited work of Teissier (1931).
The second general point is this. Just as the proportions
of a fiddler-crab are continuously changing, so may the eye-
colour of a Gammarus change throughout life, or at least
until long after sexual maturity. But a holometabolous insect
has its adult proportions fixed once for all at the attainment
of the imaginal instar ; and similarly even characters that
are determined by rate-factors will in holometabolous insects
become fixed at the same period — the final moult will take
a cross-section across the developmental curves whose results
will only be made visible at this instant. Thus a static is sub-
stituted for a dynamic picture.1
We may thus suppose that such a gradation as that presented
by the eye-colour of various multiple allelomorphic series
(e.g. of the white-eye locus in Drosophila) represents merely a
1 This is not rigidly true : there are, for instance, some eye-colours
in Drosophila which change with the age of the imago, but such changes
play a minor part as compared with those in, e.g., Gammarus, and it
is quite possible that they are not of the same nature ; in any case they
are not known to be related to size.
232 PROBLEMS OF RELATIVE GROWTH
cross-section across a series of curves for a number of similar
processes occurring at different rates, such as are actually
visible in Gammarus, but could in Drosophila only be detected
by chemical means or by deduction.
The Gammarus eye-colour curves also show the phenomenon
of an equilibrium-position — i.e. the degree of pigmentation
finally ceases to change. This merely means that pigment
and eye-area are now increasing at the same rate, instead of
the pigment-formation proceeding at a more rapid rate as at
first — in other words, it corresponds to a change from heter-
ogony to isogony, such as we have seen to occur in regard to
the size-relations of many organs (Chapter I, etc.). In the
case of rate-genes, when this change occurs before birth, we
get an apparently fixed character ; but we may either observe
(Gammarus wild-type eye) or deduce (human black eyes) that
rate-factors have been operative in the prenatal period.
With all these similarities, we may with some confidence
assume that the proportions of parts, at least in so far as they
are determined during the period of auxano-differentiation and
not during that of histo-differentiation, will be found to be
determined by rate-factors operating according to the same
rules as have been found to hold for the rate-factors controlling
pigmentation in the eye of Gammarus.
For an interesting corollary of this, we may look to the work
of Davenport (1923 ; his p. 151) on human body-build. He
has plotted the frequency for relative chest-girth (chest-girth :
stature) in man at various ages in a three-dimensional form.
If the mean values alone were taken, the age-change would
consist in a marked downward slope of the curve from birth
to late adolescence, followed by a slight upward tendency,
and then often a stable equilibrium-position. When the fre-
quency model is examined, however, it is seen not only that
there is very great variation at all ages, but that there is a
well-marked multimodality. The general shape of the curve
indicates that we are dealing with interrelated rate-factors ;
the multimodality shows that there exist a few main genetic
types, each corresponding to one sort of body-build and
determined by a particular gene or set of genes influencing
relative growth of chest-girth and/or stature (Fig. 101).
An important corollary of all this is that we should expect
numerous genetically-determined characters concerning relative
size of parts to alter with absolute size of body, without there
being any change in their genetic basis. The unusual propor-
RELATIVE GROWTH AND GENETICS
233
tions of the limbs, e.g. in human pygmies, may probably be
accounted for, wholly or in large part, along these lines.
Similarly, the general sexual differences in limb-proportions
in man, women usually having relatively short arms, and
relatively shorter legs, with in either case the distal portion
of the limb short relatively to the proximal portion (see e.g.
Hooton, 1931, p. 255) may also be due to altered time-relations
of relative growth. The curves given by Davenport (1926)
Fig. 10 1. — Body-build in Human Beings. View of a solid model, illuminated
from the top of the page, showing distribution of relative chest-girth in
man from birth to 20 years.
Ordinates, relative chest-girth (chest girth as a percentage of stature). Abscissae, age in years.
The mean relative chest-girth diminishes to about 15 years, then rises again. There are several
well-marked modes for relative chest-girth which persist throughout, indicating the presence of
distinct genetic types.
give strong confirmation to this view. See also Wallis (1932)
who finds, as is to be expected on the basis of the ideas set
forth in this volume, that relative limb-size in boys and girls
is more strongly correlated with absolute height than with age.
If the main principles involved would appear to be compara-
tively simple, the detailed complexity which may occur in
certain cases can be realized by reference to the problem of
the size of the eye and its two lobes in various members of
234 PROBLEMS OF RELATIVE GROWTH
the bar-eye series of Drosophila (for details see Goldschmidt,
1927 ; and notably Hersch, 1928).
An interesting case where genes independently control
growth-intensity in length and in breadth is afforded by the
studies of Sinnott and his associates (see Sinnott and Hammond,
1930), who find that fruit-shape in the gourd Cucurbita depends
on the interaction of various separate factors of this nature.
Sinnott and Durham (1929) find that most of the form-
differences depend primarily on the growth of the carpellary
tissue, the wall of the ovary and fruit merely following the
form-changes induced by the central region. The results of
Sinnott (1930). may be interpreted to mean that there exist
various genes controlling size and shape independently, but
that the shape-factors generally control growth-ratios, and
therefore there is a progressive change of shape with absolute
size in individual development.
An instance of a single gene having a differential but appar-
ently graded effect on a localized region is found in the house-
mouse, where the gene for short ears produces a marked short-
ening of the skull, especially in the anterior-palate region ;
a very marked diminution in the height just behind the incisors
with a slighter diminution posteriorly, and an increase in
width, rather more marked posteriorly than anteriorly (Snell,
1931).
§ 6. Relative Growth, Embryology, and Recapitulation
A rather different set of evolutionary problems from those
we considered previously has light shed on it by a considera-
tion of differential growth-coefficients and the rate-genes which
we must postulate to regulate them. These are the problems
of recapitulation and of vestigial organs.
There are numerous cases of recapitulation which cannot,
in my opinion, be accounted for along these lines, e.g. the
presence of notochord and gill-clefts in the embryos of the
highest vertebrates, but there are many others which do receive
some explanation from this source. E.g. we find that the
relatively long-armed Gibbons have a fetus which, though
longer-armed than that of other anthropoids, is relatively
shorter-armed than the adult (Schultz, 1926). Presumably
the line of least biological resistance in altering proportions of
limbs, etc., is to modify relative growth-rates, rather than to
modify the original partition of material in the early embryo
between organ and rest-of-body. And if this is so, the recapitu-
RELATIVE GROWTH AND RECAPITULATION 235
latory phenomenon follows automatically. Similar reasoning
will apply to such cases as the negative heterogony of the tail
of fetal man or positive heterogony of the toes of the Jacana
after hatching (Beebe ; see p. 262).
As regards vestigial organs, the arm-chair critic often
demands of the evolutionist how the last stages in their reduc-
tion could occur through selection, and why, if reduction has
gone as far as it has, it could not go on to total disappearance.
In the light of our knowledge of relative growth, we may
retort that we would expect the organ to be formed of normal
or only slightly reduced relative size at its first origin, but
then to be rendered vestigial in the adult by being endowed
with negative heterogony. If rate-genes are as common as
they appear to be, then what we have called the line of bio-
logical least resistance would be to produce adult vestigiality
of an organ by reducing its growth-coefficient. So long as it
is reduced to the requisite degree of insignificance at birth
(or at whatever period a larger bulk would be deleterious),
there is no need for reduction of its growth-rate to be pressed
further. But the negative heterogony with which it is endowed
will continue to operate, and it will therefore continue to
grow relatively smaller with increase of absolute size. This
last fact may account for the apparently useless degree of
reduction seen in some vestigial organs, e.g. that of the whale's
hind-limb. The degree of reduction may be useless considered
in relation to the adult, but the relative size in the adult may
be merely a secondary result of the degree of negative heter-
ogony needed to get the organ out of the way, so to speak,
before birth. In addition threshold mechanisms will possibly
be at work, so that the organ, after progressive reduction,
eventually disappears entirely.
In such cases quite small differences in growth-ratio, if the
range of absolute size over which they operate is considerable,
will make quite large differences in final relative size, a fact
which indubitably will help to account for the high variability
of vestigial organs. Even when the organ itself never grows,
as in the imaginal structures of insects with a metamorphosis,
a similar degree of variability may be brought about by
relatively small variations in the rate-genes responsible. As
Goldschmidt has shown by his classical researches on sex
in Lymantria, the end-result in such cases depends on the
amount of some effective substance produced by the gene at
the critical period for differentiation of the organ affected.
236
PROBLEMS OF RELATIVE GROWTH
Since both the rate of production of the substance and the
precise time of differentiation can be varied both by environ-
mental and genetic agencies, there is room for considerable
elasticity in the end-result provided that this is between the
normal upper and lower thresholds for the development
of the character in question. E.g., in normal male Lyman-
tria an adequate
excess of male-
determining over
female - determin-
ing substance is
present well before
the period of
differentiation;
but in female
intersexes this ex-
cess of male-deter-
mining substance
i s transformed
into an excess of
female - determin-
i n g substance
during the differ-
entiation phase.
Thus whereas
normal males are
not affected as
regards their sex-
characters by en-
vironmental agen-
cies such as
t emperature,
similar agencies
applied during the
development o f
an intersex can
exert a marked
influence on the degree of intersexuality.
So with vestigial organs (Fig. 102). If the production of a
limb-determining substance, for instance, be normally above
the necessary threshold well before the onset of the phase of
differentiation during metamorphosis, the limbs will be stable
organs, little affected in their proportions or characters by
Fig
102. — Hypothetical curves to illustrate the
relation of rate-genes to vestigial organs.
1 — 1 , lower threshold of amount of determining substance which
must be produced by the rate-gene before differentiation of the
organ can begin. 2 — 2, upper threshold ; if this is reached during
the period of differentiation, complete development of the organ
occurs. X— X, effective time of differentiation. This may be
varied so as to occur earlier (X~) or later (X + ). A — A, action of
rate-gene promoting normal complete development of organ ; the
full effective amount of determining substance is always produced
before 'the period of differentiation. B — B, action of rate-gene
whose intensity has been reduced to produce the total evolutionary
disappearance of the organ. The lower threshold of determining
substance is never reached before the period of differentiation.
C — C, action of rate-gene normally promoting a vestigial develop-
ment of the organ. Slight variations in its intensity or in the time
of differentiation will produce large differences in the degree of
development of the organ.
RELATIVE GROWTH AND RECAPITULATION 237
environmental variations. Equally, if the rate-genes con-
cerned are so much reduced in intensity that the limb-determin-
ing substances are well below the threshold during the whole
differentiation period, the absence of limbs will be a stable
character. But if the intensity of the rate-genes has only been
reduced to a pitch at which a certain small amount of limb
development can normally proceed, environmental agencies
will quite certainly be capable of affecting the precise degree
of this development within fairly wide limits. A good example
of variability of this sort in vestigial organs is provided by the
vestigial limbs of the stump-legged mayfly — Campsurus segnis,
described and figured by A. H. Morgan (1929).
A somewhat similar fact, due to somewhat similar causes,
is found in Gammarus (op. cit.). Normal black eyes, which
are due to genes whose influence is to take melanin formation
very rapidly up to saturation-point, are scarcely affected by
temperature. Other genes slow down the process so much
that the threshold for visible melanin-production is normally
never reached. The permanently pure red eyes thus produced
also are little affected by temperature. Genes of intermediate
strength, however, which at room temperature produce a
moderate darkening to red-brown, are markedly influenced
in their effects by temperature-changes : at high temperatures
the eyes of animals containing such genes are dark chocolate,
at low temperatures almost pure scarlet (Fig. 103).
Finally, since rate-genes can obviously mutate both in the
plus and the minus direction, so as to accelerate or slow down
the processes which they control, it is clear that changes in
rate-genes could as easily lead to the opposite of recapitulation
as to recapitulation. Many examples of neoteny would fall
under this head. See e.g. Mjoberg's discussion (1925) on the
larviform females of certain Lycid beetles. De Beer (1930)
has dealt fully with this point, and has proposed a useful
terminology.
Finally, it is clear that numerous examples of von Baer's
Law, that related forms tend to resemble each other more in
the early stages of their development than when adult, need
have nothing whatever to do with Haeckel's reformulation
of von Baer's Law, or with recapitulation, but are simply
consequences of the laws of relative growth and of physio-
logical genetics. For instance, the proportions of the fore-
limbs of man and all anthropoids are more similar in the fetus
than in the adult (Schultz, 1926). But in every case the
238
PROBLEMS OF RELATIVE GROWTH
relative length of the fore-limb increases between fetus and
adult. This happens to be ' recapitulatory ' in excessively
long-armed forms like the gibbon, but is the reverse of reca-
pitulatory in man, who is undoubtedly descended from brachi-
ating ancestors with relatively longer arms. The great
similarity of the fetuses is due to the fact that in all cases
the arms begin relatively small, and that final differences of
proportion are due to the different degrees of positive heter-
ogony which they exhibit. The ' recapitulation ' by the
new-born lamb in improved breeds of sheep of the approximate
proportions of adult wild sheep (Chapters III, VII) is another
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Fig. 103. — Effect of temperature on eye-darkening in a pure genetic
strain (rrSS) of Gammarns chevreuxi.
At io° C. no melanin is deposited. At 13°, deposition begins only after 5-6 months. The figure
shows the facts for temperatures from 150 to 28°.
example of a change of this sort which happens to be reca-
pitulatory.
All coloured Gammarus eyes so far investigated begin as
pure red (Ford and Huxley, op. cit.). But we have in this
no grounds whatever for supposing that Gammarus are de-
scended from a red-eyed ancestor. It would appear to be a
direct consequence of the empirical fact that melanin deposi-
tion can only begin after the red (lipochrome) pigment has
been formed.
Again, the fact that both chelae of male Uca begin of female
type, so that males and females are more alike when young,
does not in the least prove that the males of any ancestral
Uca ever possessed two chelae of existing or any other female
type. In fact, all the probabilities are against this. It is due,
RELATIVE GROWTH AND RECAPITULATION 239
we may suggest, rather to the fact that a large adult chela
must begin its existence of small relative size, and that the
form of the small chela has been moulded by natural selection
into the female-type or feeding chela. A similar reasoning
applies to the male type of juvenile female abdomen in all
Brachyura : this affords no presumption whatever that
ancestral female crabs when adult ever possessed an abdomen
as narrow as that of the existing male. The presumption is
quite different — that during Brachyuran evolution the male
abdomen has been narrowed, the female broadened, and that
it has proved an ontogenetic convenience to produce the
abdomen at its first appearance, when it is relatively small,
in a form similar to that of the male, from which the female
type is derived by differential heterogony.
Numerous other examples could be given of this distinction
between von Baer's and Haeckel's laws ; but enough has been
said to show the importance of the distinction, and the reason
for the greater universality of von Baer's generalization.
Further, it is in general clear that rate-genes may mutate
in either a plus or a minus direction, either accelerating or
retarding the rates of the processes they affect. In the former
case the effect will be in certain respects at least recapitulatory,
since a condition which used to occur in the adult is now run
through at an earlier stage. In the latter case, the effect
will be anti-recapitulatory, since a condition which once
characterized an earlier phase of development is now shifted
to the adult phase. Neoteny is the most striking example
of this effect ; but we may also get single characters behaving
in this way. When this occurs, previous adult characters,
though still in a real sense potentially present, never appear
because their formation is too long delayed : they are lost
to the species by being driven off the time-scale of its develop-
ment. For examples of this, notably in Helix, see the discus-
sion in Ford and Huxley, 1927. (Fig. 104.)
De Beer (1930), in his resume of this and kindred subjects,
has given the useful term paedomorphosis to this type of effect.
Bolk (1926) has drawn attention to the importance of such
effects in the evolution of man, summing up the result as a
process of ' fetalization ', since in many respects post-natal
or adult man resembles the fetal stages of apes. See also
Kieslinger (1924).
Such phenomena are unintelligible on the Haeckelian
doctrine. But they immediately fall into place when it is
240
PROBLEMS OF RELATIVE GROWTH
grasped that most examples of recapitulation constitute
simply one side of a more general problem — the problem of
altering the relative rates of growth and of other processes
within the body.
t
t
S
I1
X ^"n
y
s
s _
,-'l
A
/ ,'''
\ y
^-JK
. **
sr>
£ — ^r
• (a)
/j/
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fA"
/ / / "
// /j
X
time
Fig. 104. — Diagram to illustrate positive and negative mutations in rate-
factors, leading to recapitulation and paedomorphosis respectively.
X — X, period after which no further differentiation of the character is possible (adult phase in
forms with fixed adult size ; maximum size in forms with continuous growth). The curves represent
the development of a character controlled by a rate-gene. The portions of the curves to the right
of X — X (dotted) cannot be realized. I, condition in ancestral form; II, result of a mutation
accelerating the process. The old adult condition (A) now becomes the juvenile condition (J').
A portion of the curve hitherto unrealized is now incorporated in the life-history (heavy line) leading
to a new adult condition (A'). Ill, result of a mutation slowing down the process. The old juvenile
condition (J) now becomes the adult condition (A"). A portion of the curve (heavy dotted line),
including the old adult condition (a), is now extruded from the life-cycle by retardation.
§ 7. General Approach to the Problem of Qualitative
Form-change
In general, it is clear that the effect of our knowledge of rate-
factors and of the rules regulating relative growth will make
it possible to analyse the problems involved in changes in
proportion with a new insight.
In studying the growth of any part, it is by no means suffi-
cient to determine its percentage change in relative size, as
has often been the case in the past. A percentage change is
the easiest to visualize by graphic methods, and is often very
important in comparing related forms. But from the point
QUALITATIVE FORM-CHANGE 241
of view of the underlying mechanism, it tells us little. A
percentage change in relative size, as the work of Scammon
has made clear (p. 132), may be merely an effect of the law
of developmental direction and the time-handicap thus given
to some organs as against others ; the organ, once it begins
its growth, may grow in linear proportions to the body as a
whole, and yet show a change in relative size. Or a per-
centage change may be due solely to the heterogony of the
organ, to its possessing a growth-coefficient above or below
the body as a whole. Or thirdly, it may of course be and
doubtless often is due to a combination of the two causes.
In the first case, its relative growth will be according to the
formula
y = bx + a (1)
In the second case, according to the formula
y = bxk (2)
In the third case, according to the formula
y = bxk + a (3)
When k = 1, then formula (1) becomes equivalent to formula
(3). And when in addition a = o, it becomes equivalent to
formula (2). Thus for those numerous cases where the growth-
coemcient is close to unity, and the onset of development of
the organ is not far from that of the standard part taken as
representative of the rest of body (or from the mean value
for all parts, when the body as a whole is taken as standard),
the three formulae will serve equally well.
Formula (3) is the most inclusive, and should be taken as
the theoretical basis for analysis ; but a will often be negligible
where heterogony is marked ; and where heterogony is not
marked, k will be so close to unity that Scammon's formula (1)
will approximately apply. (See also Chap. IV, § 7.)
In addition to a precise analysis of growth after the organ
has appeared, it is also of theoretical importance to know
the time of its first appearance. From the work on Gammarus,
we know that, as regards the straightforward rate-genes
discussed in § 5, the slower the rate of the process which they
control, the later is its visible onset.
This is presumably due to there being a maximum threshold
which must be attained by the substance whose production
they control before visible effects are produced. It is on
general grounds probable that this is a widespread rule. In
addition, genes have been discovered whose primary effect
16
242 PROBLEMS OF RELATIVE GROWTH
is on the time of onset of a process, and not on its rate (Ford
and Huxley, op. cit.).
It should be possible by comparing related forms to discover
whether time-relations of this second type are involved in
addition to those of the first. We must further remember
that growth-coefficients may change during ontogeny ; they
may change from strong to less strong positive heterogony, as
in Uca large chela, or from isogony to positive heterogony,
as in Maia large chela, etc. These facts must also have their
genetic basis.
With an analysis such as this, we may hope for a fuller
understanding of the processes involved in changes of propor-
tion. One alteration in a single rate-gene may delay the first
formation of an organ and also decrease the growth-coefficient
once it is formed. Further, although the processes of histo-
differentiation do not seem to follow the same laws of relative
growth as those of auxano-differentiation, the quantitative
intensity of the two kinds of growth-processes may well be
controlled by the same genes.
In considering evolutionary changes in relative size, we
must accordingly try to distinguish the various agencies which
may be at work. It appears that these may be (a) mutations
affecting the primary gradient of the early embryo, on which
the time-relations of antero-posterior differentiation depend ;
(b) mutations affecting specific rate-genes ; (c) mutations
affecting specific ' time-genes ' — genes controlling time of onset
and not rate of processes. The processes controlled by the
rate-genes and time-genes will be processes concerned with
growth-gradients, whether of a major or minor nature : they
will therefore always affect a number of parts in a correlated
way.
Finally, it is at least possible, as we have seen in an earlier
section, that the primary ' axial gradient ' of the developing
egg and early embryo, on which, we must suppose, depend
the facts subsumed under the law of antero-posterior differ-
entiation, itself continues to operate later as a growth-influenc-
ing gradient, as well as influencing the time-relations of
differentiation. This would mean that mutations primarily
selected because of their effect upon early development would
have an effect upon proportional size in later life — an interest-
ing example, if substantiated, of what Darwin called correlated
variation.
CONCLUSION 243
§ 8. Conclusion
We have now completed our brief survey. Starting from
the fact of obviously ' dysharmonic ' or heterogenic growth,
we have discovered our first new empirical law — the law of
constant differential growth-ratio. We have then recognized
that it is only a special case of the law of differential growth-
partition, which is the prime quantitative basis of relative
growth. Passing on from that, we have found a further and
quite unexpected empirical law — that the existence of a
differential growth-ratio in an organ or region seems always
to be associated with a growth-gradient culminating in a
growth-centre ; or in other words, that the distribution of
growth-potential is not marked by discontinuities or by
frequent oscillations, but occurs in an orderly and continuously
graded way. And we then showed that these localized growth-
gradients were but special cases of growth-gradients permeating
the whole body. These laws, however, only appear to apply
to the stages of growth occurring after histological differentia-
tion has been completed. Very rapid growth, obeying quite
other laws, occurs during the earlier period. For these two
phases of development, the terms histo-differentiation and
auxano-differentiation are proposed.
After demonstrating that these growth-gradients were opera-
tive both in multiplicative and accretionary growth, giving
rise to structures as dissimilar as a crustacean chela or a
fowl's comb on the one hand, and a Nautilus shell or a rhino-
ceros horn on the other, we made it probable that the growth-
gradients were either directly or indirectly correlated with
the morphogenetic gradients or fields of Child, Weiss and others,
and in general with the various polarized and field effects in
the animal body.
In a discussion of the obscure subject of the physiological
basis of growth-gradients, we discovered that the existence
of a single appendage with high growth-ratio is associated
with a slight increase of growth in the regions immediately
posterior to it, but a slight decrease in those immediately
anterior. The meaning of this remains quite unknown, but
it has certain parallels in the field of regeneration and of
experimental embryology. Further, the study of relative
growth confirms that of regeneration in making us believe
that the relative growth-rate (differential growth-ratio) of a
part is determined in some way as an equilibrium between
244 PROBLEMS OF RELATIVE GROWTH
the growth of the part and the growth of the rest of the body.
The role of hormones and of mutation in differential growth
has been discussed, and the extent of our ignorance on this
subject emphasized.
Finally, the bearings of the study of differential growth on
other branches of biology have been discussed, and it has been
shown that light is thereby shed upon such diverse problems
as orthogenesis, recapitulation, vestigial organs, the existence
of non-adaptive characters, physiological genetics, comparative
physiology, and systematics.
I may conclude as I began, by a quotation from D'Arcy
Thompson, to whose classical work all students of relative
growth owe so much [Growth and Form, p. 719) :
" The study of form may be descriptive merely, or it may become
analytical. We begin by describing the shape of an object in the
simple words of common speech : we end by defining it in the precise
language of mathematics ; and the one method tends to follow the
other in strict scientific order and historical continuity".
BIBLIOGRAPHY
Abeloos, M. (1928) : Sur la dysharmonie de croissance chez Planaria
gonocephala Duges et sa reversibilite au cours du jeune ; C.R.
Soc Biol., 98, 917.
(1930) : Recherches experimentales sur la croissance et la re-
generation chez les Planaires ; Bull. Biol. France et Belg., 64,
1-140.
Adolph, E. F. (1930) : Body Size as a Factor in the Metamorphosis
of Frogs ; Anat. Rec, 47, 304.
Allen, B. M. (191 8) : The Results of Thyroid Removal in the Larvae
of Rana pipiens ; Journ. Exper. Zool., 24, 489-519.
(191 9) : The Development of the Thyroid Glands of Bufo and
their normal relation to Metamorphosis ; Journ. Morph., 32,
489-507.
Alpatov, W. W. (1930) : Phenotypical Variation in Body and Cell-
size of Drosophila melanogaster ; Biol. Bull., 58, 85-103.
Anderson, B. G. (1931) : Relative Growth in Daphnia Magna ; Anat.
Rec, 51, Suppl., p. 55.
Antonius, H. O. (1920) : Bemerkung iiber einige Saugetierschadel
von Sardinien ; Kon. Akad. Wetens. Amsterdam, 29, 26, vi. 20,
P- 254.
Appleton, A. B. (1929) : The Relation between the Rate of Growth
and the Rate of Ossification in the Foetus of the Rabbit (pre-
liminary communication) ; Proc. Assoc. Anat., 1929, p. 23.
Atkins, D. (1926) : The Moulting Stages of the Pea-crab {Pinnotheres
pisum) ; Journ. Mar. Biol. Assn., 14, 475.
Balfour-Browne, F. (1909) : The Life-history of the Agrionid
Dragonfly ; Proc. Zool. Soc. Lond., 1909, 253-84.
Bateson, W. (1894) : Materials for the Study of Variation. London :
Macmillan.
Bateson, W. and Brindley, H. H. (1892) : On some Cases of Varia-
tion in Secondary Sexual Characters statistically examined ;
Proc. Zool. Soc. (1892), 585.
Bean, R. B. (1924) : The Pulse of Growth in Man. A preliminary
report ; Anat. Rec, 28, 45-61.
Becher, A. (1923) : (Sheep-dog Skull). Arch. Naturgesch. (A.) 1923, (9).
de Beer, G. R. (1930) : Embryology and Evolution. Oxford :
Clarendon Press.
Belehradek, J. and Huxley, J. S. (1930) : The Rate of Eye-growth
and its Variation in Gammarus chevreuxi ; Journ. Exper. Biol.,
7, 37-
Benazzi, M. (1929) : Manifestazioni quantitative della rigenerazione
negli insetti ; Rivista di Biologia, 1 1 , fasc. 5-6, p. 9.
245
246 PROBLEMS OF RELATIVE GROWTH
Benoit, J. (1927A) : Obtention chez les coqs d'un etat intermediate
de la crete . . . ; C.R. Soc. Biol., 97, 275.
(1927B) : La croissance conditionnee de la crete . . . ; C.R. Soc.
Biol., 97, 279.
Berkson, J. (1929) : Growth Changes in Physical Correlation — height,
weight and chest-circumference, males; Human Biol., 1,
462-502.
Berrill, N. J. (1931) : Regeneration in Sabella pavonina (Sav.) and
other Sabellid Worms ; Journ. Exper. Zool., 58, 495-523.
Bertalanffy, Ludwig (1928) : Kritische Theorie der Formbildung.
Berlin : Borntraeger.
Blyth, J. S. S., Dodds, E. C. and Gallimore, E. J. (1931) : Observa-
tions on the Assay of the Comb-growth Promoting Hormone ;
Journ. Physiol., 73, 136-40.
Bolk, L. (1926) : Das Problem der Menschwerdung. Jena, 1926.
Bo vet, D. (1930) : Les territoires de regeneration ; leurs proprietes
etudiees par la methode de deviation du nerf ; Rev. Suisse Zool.,
37, 83-145.
Bower, F. O. (1930) : Size and Form in Plants, with special reference
to the Primary Conducting Tracts. London : Macmillan.
Bray, G. (1931) : Recent Advances in Allergy. London : Churchill,
1931-
Brody, S. (1928) : An Analysis of the Course of Growth and Senescence.
In Growth, by Robbins and other authors. Yale University
Press.
Bush, S. F. (1930) : Asymmetry and Relative Growth of Parts in the
two Sexes of the Hermit-crab Eupagurus prideauxi ; Arch.
Entw. Mech., 123, 39.
Bush, S. F. and Huxley, J. S. (1930) : Distribution of Growth-activity
in Eupagurus ; Nature, 126, 240-1.
Calman, W. T. (1909) : Treatise on Zoology, Vol. VII. Appendiculata,
3rd fasc. Crustacea. London : Black.
Champy, C. (1922) : (Thyroid) ; Arch. Morph. Gen. Exp., 1922.
(1924) : Sexualite et Hormones. Paris : G. Doin.
(1929) : La croissance dysharmonique des caracteres sexuels
accessoires ; Ann. Sci. Nat. Zool., (10) 12, 193.
Child, C. M. (1915) : Senescence and Rejuvenation. Chicago.
Choi, M. H. (1931) : Synchronism of Development in Parabiotic
Amphibian Larvae ; Folia Anatomica Japonica, 9. No. 5.
Clark, A. (1927) : Comparative Physiology of the Heart. Cambridge
Univ. Press.
Clausen, H. J. (1929) : Rate of Histolysis of Anuran Tail Skin and
Muscle during Metamorphosis ; Anat. Rec, 44, 218.
Coghill, G. E. (1928) : VIII. (Growth of Nervous System) ; Journ.
Comp. Neurol., 45.
Collip, J. B. (1930) : (Placental Hormones) ; Canad. Med. Assoc.
Journ., 23, 631.
Cott, H. B. (1929) : Observations on the Natural History of the
Racing-crab Ocypoda ceratophthalma, from Beira ; Proc. Zool.
Soc. (1929), 755-
Cushing, H. (1912) : The Pituitary Body and its Disorders ; Lippincott.
BIBLIOGRAPHY 247
Davenport, C. B. (1923) : Body-build and its Inheritance ; Carnegie
Inst., Washington, Publ., No. 329, p. 176.
Davenport, C. B. (1926) : Human Metamorphosis ; Amer. Journ. Phys.
Anthrop., 9, 205-32.
Davenport, C. B. and Swingle, W. W. (1927) : Effects of Operations
upon the Thyroid Glands of Female Mice on the Growth of
their Offspring ; Journ. Exper. Zool., 48, 395-440.
Detwiler, S. R. (1930) : Some Observations upon the Growth,
Innervation and Function of Heteroplastic Limbs ; Journ.
Exper. Zool., 57, 183.
Diakonov, D. M. (1925) : Experimental and Biometrical Investiga-
tions on Dimorphic Variability of Forficula ; Journ. Genetics,
15, 261.
Dombrowski, von (1889) : TJber die Geweihbildung der Rothirsche
der Gegenwart . . . ; Der Weidmann, 20 (Nos. 15-19, 42, 44).
(1892) : Die Gehornbildung des Rehbockes des Gegenwart, etc. ;
Der Weidmann (1892-4), 22, 23, 24.
Domm, L. V. and Juhn, Mary (1927) : Compensatory Hypertrophy
of the Testes in Brown Leghorns ; Biol. Bull., 52, 458-473.
Donaldson, H. H. (1924) : The Rat ; Philadelphia : Wistar Inst. Mem.
Drennan, M. R. (1932) : Pedomorphism in the Pre-Bushman Skull ;
Amer. Journ. Phys. Anthrop. (cited from Wistar Inst. Advance
Abstract-sheet No. 179).
Dubois, E. (1914) : Die Gesetzmassige Beziehungvon Gehirn- zur Kor-
pergrosse bei den Wirbeltieren ; Zeits. Morph. u. Anthrop., 18.
(191 8) : On the Relation between the quantities of the Brain,
the Neurone and its Parts, and the Size of the Body ; Acad.
Sci. Amsterdam, Proceedings, 20, 1328.
(1922) : Phylogenetic and Ontogenetic Increase of the Volumes
of the Brain in Vertebrata ; K. A had. Wetens. Amsterdam, 25,
330-
Dudich, E. (1923) : Ueber die Variation des Cyclommatus tarandus
Thunberg ; Arch. f. Naturgesch., (A) 1923, (2) 62.
Eigenbrodt, H. J. (1930) : The Somatic Effects of Temperature on
a Homozygous Race of Drosophila ; Physiol. Zool., 3, 392-411.
Emerson, A. E. (1926) : Development of a Soldier of Nasutitermes
(Constrictotermes) . . . ; Zoologica, 7, 69.
Emery, C. (1921) : Quels sont les facteurs du Polymorphisme du sexe
feminin chez les Fourmis ? ; Rev. Gen. Sci., 32.
Entz, G. (1927) : Beitrage zur Kenntnis der Peridineen. II. resp. VII.
Studien an Siisswasser-Ceratien ; Arch. f. Protistenk., 58,
344-440.
Faure-Fremiet, E. (1930) : Growth and Differentiation of the colonies
of Zoothamnium alternans ; Biol. Bull., 58, 28.
Ford, E. (1930) : Herring Investigations at Plymouth. VIII. The
transition from larva to adolescent ; Journ. Mar. Biol. Assn.,
16, 723-
(193 1 a) : Growth in Length during the Transition from Larva
to Adolescent in the Pilchard and Sprat : Journ. Mar. Biol.
Assn., 17. 977-85.
248 PROBLEMS OF RELATIVE GROWTH
Ford, E. (1931B) : Changes in Length during the Larval Life and
Metamorphosis of the Freshwater Eel (Anguilla vulgaris Turt.) ;
Journ. Mar. Biol. Assn., 17. 987-1000.
Ford, E. B. (1928) : The Inheritance of Dwarfing in Gammarus chev-
reuxi ; Journ. Genet., 20, 43.
(1929) : The Physiology of Genetics; Eugen. Rev., 21, 114.
Ford, E. B. and Huxley, J. S. (1929) : Genetic rate-factors in Gam-
marus ; Arch. Entw. Mech., 117, 67.
Gabritschevsky, E. (1930) : Les reductions regulatrices et les com-
pensations hypertrophiques pendant l'ontogenese et la regenera-
tion de l'araignee Thomisus onistus ; Bull. Biol., 64, 155-88.
Gajewski, N. (1922) : Ueber die Variabilitat bei Artemia salina ;
Intern. Rev. Ges. Hydrobiol., 10, 139, 299.
Gause, C. F. (1931) : Ueber den Einfluss verkiirzter larvaler Ernahr-
ungszeit auf die Eiergrosse von Drosophila ; Biol. Zentralbl.,
51, 209.
Giesbrecht, W. (1910) : Fauna und Flora des Golfes von Neapel ;
33, Stomatopoda (i)
Goldschmidt, R. (1923) : The Mechanism and Physiology of Sex
Determination ; London : Methuen.
(1927): Physiologische Theorie der Vererbung ; Berlin: Springer.
Gray, J. (1929) : The Kinetics of Growth ; Brit. Journ. Exper. Biol.,
6, 248.
Griffini, A. (191 2) : Strane variazioni individuali in alcune specie di
Coleopteri ; Boll. Mat. Sci. Fis. Nat., Lodi, 13, (8).
Guyenot, E. and Ponse, K. (1930) : Territoires de regeneration et
transplantations ; Bull. Biol. Fr. et Belg., 64, 251-84.
Hammett, F. S. (1929A) : Thyroid and Differential Development ;
Endokrinologie , 5, 81.
(1929B) : Thyroid and Growth ; Quart. Rev. Biol., 4, 353.
Hammond, J. (1921) : On the Relative Growth and Development of
Various Breeds and Crosses of Sheep ; Journ. Agric. Sci., 11,
267-407.
(1928) : Selection for Meat Production ; Verh. v. Internal. Kongr.
Vererbungswiss, Berlin, 1927, 2, 769-95.
(1928) : How Feeding affects Conformation ; Farmer and Stock-
breeder and Agric. Gaz., 10.12.28, p. 4.
(1929) : Probleme der Fleischerzeugung ; Zeits. Zuchtungskunde,
4, 543-62.
(1930) : Animal Production : the Influence of Natural Conditions
on Animal Production ; Trop. Agric, 7, 147-50.
(1931) : Growth of the Rabbit in Relation to Flesh Production ;
Year Book Nat. Rabbit Council (1931).
Hardest y, Mary (1931) : The Structural Basis for the Response
Comb of the Brown Leghorn Fowl to the Sex Hormones ; Amer.
Journ. Anat., 47, 277-323.
Hare, F. (1932) : Polymorphism among the Sub-genera of Nasuti-
termes ; Journ. Morph. and Physiol, (in press).
Harris, H. A. (1931A) : The Comparative Aspect of Growth in Children ;
Lancet, 28 Mar., 1931, p. 691.
BIBLIOGRAPHY 249
Harris, (1931B) : The Anatomical and Physiological Characteristics and
Development of Children between the ages of 7 and 1 1 ; Report.
Consultative Committee on the Primary School, pp. 222-54.
Harrison, R. G. (1924) : Some unexpected Results of the Hetero-
plastic Transplantation of Limbs ; Proc. Nat. Acad. Sci., 10,
69-74.
R. (1929A) : Heterotransplantation in Amphibian Embryos ; Proc.
X. Internal. Zool. Congr. Budapest, 1927 (1929).
(1929B) : Correlation in the Development and Growth of the
Eye studied by means of Heteroplastic Transplantation ; Arch.
Entw. Mech., 120, 1.
Hasebrock, K. (1917) : Die Entwicklungsmechanik des Herzwachs-
tum ; Pfluger's Archiv, 168.
Haseman, J. D. (1907A) : The Direction of Differentiation in regenerat-
ing Crustacean appendages: Arch. Entw. Mech., 24, 617.
(1907B) : The Reversal of the Direction of Differentiation in the
Chelipeds of the Hermit-crab ; Arch. Entw. Mech., 24, 663.
Heath, H. (1927) : Caste-formation in the Termite genus Termopsis ;
Journ. Morph. and Phys., 43, 387.
(1928) : Fertile Termite Soldiers ; Biol. Bull., 54, 324.
Hecht, S. (1916) : Form and Growth in Fishes ; Journ. Morph., 37,
379-
Henderson, J. R. and Matthai, G. (1910) : On certain species of
Palaemon from South India ; Records of the Indian Museum, 5,
277-306.
Herms, W. B. (1928) : The Effect of Different Quantities of Food
during the Larval period on the sex-ratio and size of Lucilia
sericata Meigen and Theobaldia incidens (Thorn) ; Journ. Econ.
Entom., 21, 720.
Herrick, F. H. (191 1) : Natural History of the American Lobster;
Bull. U.S. Bureau Fisheries, 29, 149, Document No. 747.
Hersh, A. H. (1928) : Organic Correlation and its Modification in
the bar series of Drosophila ; Journ. Exper. Zool., 50, 239.
Hesse, R. (1927) : Uber Grenzen des Wachtums ; Jena, G. Fischer,
1927.
Hickling, C. F. (1930) : A Contribution towards the Life-history of
the Spur-dog ; /. Mar. Biol. Ass., 16, 529.
Hinton, M. A. C. (1926) : Monograph of the Voles and Lemmings
(Microtinae) ; London, British Museum (Natural History).
Hirsch, G. C. (1931) : The Theory of Fields of Restitution with special
Reference to the Phenomena of Secretion ; Biol. Rev., 6, 88.
Hooton, E. A. (1931) : Up from the Ape. London : Allen & Unwin.
Hubbs, Carl L. (1925) : The Metamorphosis of the Californian Ribbon
Fish, Trachypterus rex-salmonorum ; Papers Michigan Acad. Sci.,
Arts and Letters, 5, 469-76.
Hutt, F. B. (1929) : Sex Dimorphism and Variability in the Appen-
dicular Skeleton of the Leghorn Fowl ; Poultry Science, 8,
202-18.
Huntsman, A. G. (192 1) : The Effect of Light on Growth in the
Mussel : Proc. Roy. Soc. Canada, 15, (5), 23-8.
Huxley, J. S. (1921) : Studies in Dedifferentiation. II. Dediffer-
entiation and Resorption in Perophora ; Q. J . Micr. Soc, 65,643.
250 PROBLEMS OF RELATIVE GROWTH
Huxley, J. S. (1924A) : The Variation in the Width of the Abdomen
in Immature Fiddler-crabs considered in relation to its relative
Growth-rate ; Amer. Nat., 58, 468.
(1924B) : Constant Differential Growth-ratios and their Signifi-
cance ; Nature, 114, 895.
(1924c): Early Embryonic Differentiation; Nature, 114, 3.
(1926) : The Annual Increment of the Antlers of the Red Deer
(Cervus elaphus) ; Proc. Zool. Soc. (1926), 102 1.
(1927A) : Further Work on Heterogonic Growth; Biol. Zentralbl.,
47, 151.
(1927B) : Discontinuous Variation and Heterogony in Forficula ;
Joum. Genetics, 17, 309.
(1927c) : On the Relation between Egg- weight and Body- weight
in Birds ; Joum. Linn. Soc. (Zool.), 36, 457.
(1927D) : Studies on Heterogonic Growth (IV). The bimodal
cephalic horn of Xylotrupes gideon ; Joum. Genetics, 17, 45.
(193 1 a) : The Relative Size of Antlers in Deer ; Proc. Zool. Soc.
(1931), (819-64).
(1931B) : Notes on Differential Growth ; Amer. Nat., 65, 289-315.
(1931c) : Relative Growth of Mandibles in Stag-beetles (Lucani-
dae) ; Linn. Soc. Joum. (Zool.), Vol. 37 : 675-703.
Huxley, J. S. and Callow, F. S. (unpublished) : Note on the Asym-
metry of male Fiddler-crabs (Oca pugilator).
Huxley, J. S. and Fulton, J. F. (1924) : The Influence of Temperature
on the Action of Insulin ; Nature, 16.2.24, 234.
Huxley, J. S. and Richards, O. W. (1931) : Studies in Heterogonic
Growth (8). Changes in proportions in the abdomen of the
shore-crab Carcinus maenas ; Joum. Mar. Biol. Assn., 17, 1001.
Imms, A. D. (1920) : On the Structure and Biology of Archotermopsis ;
Phil. Trans. Roy. Soc. B., 209, 75.
Jackson, C. M. (1925) : The Effects of Inanition and Malnutrition
upon Growth and Structure ; London and Philadelphia, 1925.
Jucci, C. (1924) : II polimorfismo unisessuale di un Acaro plumicolo ;
Boll. Lab. Zool. Gen. Agr. Portici., 18, 79.
Juhn, M., Faulkner, O. H., and Gustavsen, R. G. (1931) : The
Correlation of Rates of Growth and Hormone Threshold in the
Feathers of Fowls ; Joum. Exper. Zool., 58, 69.
Kalshoven, L. G. (1930) : De Biologie van de Djatitermiet Kalotermis
teitenae. . . . (H. Veenman & Zonen, Wageningen).
Keith, A. (1923) : The Adaptational Machinery concerned in the
Evolution of Man's Body ; Nature, 18-8-1923, Suppl.
Kemp, S. (191 3) : Crustacea Decapoda. Records of the Indian Museum,
8 (1912 to 1922), 289-310.
(1914) : Records of the Indian Museum, 10, 81.
(1915) '■ Fauna of the Chilka Lake, Crustacea Decapoda, Memoirs
of the Indian Museum, 5, 201—325.
Keys, Ancel B. (1928) : The Weight-length Relation in Fishes ;
Proc. Nat. Acad. Sci., 14, 922-5.
Kieslinger, A. (1924) : Neotenie, Persistenz, Degeneration ; Proc.
Konink. Akad, Wetens. Amsterdam, 27, 761-71.
BIBLIOGRAPHY 251
Klatt, B. (1913) : Ueber den Einfluss der Gesamtgrosse auf das
Schadelbild, etc. ; Arch. Entw. Mech., 36, 387-471.
(191 9) : Zur methodik vergleichender metrischer Untersuchungen,
besonders des Herzgewichtes ; Biol. Zentralbl., 39, 406.
Kozelka, A. W. (1930) : Integumental Grafting as a Means of Analysing
the Factors determining the Secondary Sexual Characters of
the Domestic Fowl ; Anat. Rec, 47, 302.
Kuhl, W. (1928) : Die Variabilitat der abdominal Korperanhange bei
Forficula auricularia ; Zts. Morph. Oekol. d. Tiere, 12, 299.
Kunkel, B. W. and Robertson, J. A. (1928) : Contributions to the
Study of Relative Growth of Parts in Gammarus chevreuxi ;
Joitrn. Alar. Biol. Assn., 15, 655.
Lameere, A. (1904) : L'e volution des ornements sexuels ; Bull. Acad.
Belg. (1904), 1337.
(191 5) : Les caracteres sexuels des Prionides ; Bull. Set. Fr. et
Belg., 49, 1915-
Landauer, W. (1927) : Untersuchungen iiber Chondrodystrophie. I.
Allgemeine Erscheinungen und Skelett chrondrodystrophischer
Hiihnerembryonen ; Arch. Entw. Mech., 110, 195-278.
■ (1929) : Thyrogenous Dwarfism {Myxcedema infantilis) in the
Domestic Fowl; Amer. Journ. Anat., 43, 1-20.
Lapicque, L. (1907) : Le poids encephalique en fonction du poids
corporal entre individues d'une meme espece ; Bull. Mem. Soc.
Anthrop., Paris (5), 8, 313.
(1922) : Le poids du cerveau et l'intelligence. Chap. 2 in ' Traite
de Psychologie ', ed. Dumas, Paris.
Lapicque, L. and Giroud, A. (1923) : Sur le nombre des fibres ner-
veuses peripheriques en fonction de la grandeur du corps ; C.R.
Soc. Biol., 88, 43.
Latimer, Homer B. and Aikman, J. M. (1931) : The Pre-natal Growth
of the Cat. I. The growth in weight of the head, trunk, fore-
limbs, and hind-limbs ; Anat. Rec, 48, 1-26.
Light, S. F. (1927) : A new and more exact Method of expressing
important specific Characters of Termites ; Univ. Cal. Publ.
Entomol., 4, 75-88.
Lillie, F. R. and Juhn, M. (1932) : The Physiology of Development
of Feathers ; Physiol. Zool., cited from Wistar Inst. Advance
Abstract-sheet.
Lund, E. J. (1923) : Threshold Densities of the Electric Current for
Inhibition and Orientation of Growth in Obelia ; Proc. Soc.
Exper. Biol, and Med., 21, 127-8.
(1928) : Relation between Continuous Bioelectric Currents and
Cell Respiration. II. (a) A theory of continuous bio-electric
currents and electric polarity of cells. II. (b) Theory of cell
correlation; Journ. Exper. Zool., 51, 265-90.
Mackintosh, N. A. and Wheeler, J. F. G. (1929) : Southern Blue
and Fin Whales ; Discovery Reports, 1 , 257-540.
Martin, R. (1928) : Lehrbuch der Anthropologic 2nd Ed. Jena,
1928.
252 PROBLEMS OF RELATIVE GROWTH
Matthew, W. D. (1926) : The Evolution of the Horse ; Quart. Rev.
Biol., 1, 139.
Mead, C. H. (1930) : A Quantitative Study in Human Teratology ;
Human Biol., 2, 1.
Mjoberg, Eric (1925) : The Mystery of the so-called ' Trilobite
Larvae ' definitely solved ; Psyche, 32, 119-54.
Morgan, A. H. (1929) : The Mating Flight and the Vestigial Struc-
tures of the Stump-legged Mayfly, Campsurus segnis Needham ;
Ann. Entomol. Soc. Amer., 22, 61-8.
Morgan, T. H. (1923A) : The Development of Asymmetry in the
Fiddler-crab; Amer. Nat., bl , 269.
(1923B) : Further Evidence on Variation in the Width of the
Abdomen in Immature Fiddler-crabs ; Amer. Nat., 57, 274.
(1924) :
Nanagas, J. C. (1925) : A Comparison of the Growth of the Body
Dimensions of Anencephalic Human Fetuses with normal Fetal
Growth as determined by Graphic Analysis and Empirical
Formulae ; Amer. Journ. Anat., 35, 455.
Needham, J. (1931) : Chemical Embryology. Cambridge. Univ. Press.
3 vols.
Nevalonnyi, M. and Podhradsky, J. (1930) : Der Einfluss der
Schilddriise und des Thymus auf die Skelettbildung . . . ; Vest-
nik Ceskoslov. Akad. Zemedelske., 6, 1.
Nomura, E. (1928) : On the Relation between Weight and Dimensions
in the Bivalves, Tapes philippinarum and Cytherea meretrix ;
Sci. Rep. Tohoku Imp. Univ. (4th Ser., Biol.), 3, 113-24.
Nomura, E. and Sasaki, K. (1928) : On the Relation between Weight
and Dimensions in the Gastropods, Haliotis gigantea var. discus
and Littorina sitchana ; Sci. Rep. Tohoku Imp. Univ. (4th Ser.,
Biol.), 3, 125.
Olmsted, J. M. D. and Baumberger, J. P. (1923) : Form and Growth
of Grapsoid Crabs ; /. Morph., 38, 279.
Osborn, H. F. (1929) : The Titanopheres of Ancient Wyoming, Dakota,
and Nebraska ; U.S. Dept. of the Interior Geol. Survey, Mono-
graph No. 55.
Palenitschko, Z. G. (1927) : Zur vergleichenden Variabilitat der
Arten und Kasten bei den Ameisen ; Zeits. Morph. u. Okol.,
9, 410.
Parkes, A. S. (1929) : The Internal Secretions of the Ovary. London :
Longmans Green & Co.
Parrot, C. (1894) : Uber die Grossenverhaltnisse des Herzens bei
Vogeln ; Zool. Jahrb. (Syst.), 7.
Parsons, F. G. (1927) : The Englishman of the Future ; Rept. Brit.
Assoc. Leeds, 1927, Pres. Address, Section H.
Pearsall, W. H. (1926) : Growth Studies. V. Factors affecting the
development and form of leaves ; Ann. Bot., 40, 85-103.
■ (1927) : Growth Studies. VI. On the relative sizes of growing
plant organs; Ann. Bot., 41, 549.
BIBLIOGRAPHY 253
Pearsall, W. H. and Hanby, A. M. (1925) : The Variation of Leaf
Form in Potamogeton perfoliatus ; New Phyt., 24, 112-20.
Pearson, J. (1908) : L.M.B.C. Memoir No. 16 ; Cancer. London :
Williams & Norgate.
Perkins, M. (1929) : Growth Gradients and the Axial Gradients of
the Animal Body ; Nature, 124, 299.
Petersen, Chr. (1921) : Das Quotientengesetz : eine biologisch-
statistische Untersuchung ; Kopenhagen, Bianco Lunos, 1921.
Pezard, A. (191 8) : Le conditionnement physiologiques des carac-
teres sexuels secondaires chez les Oiseaux.
(1921) : Numerical Law of Regression of certain Secondary Sex
Characters ; Journ. Gen. Physiol., 3, 271.
Przibram, H. (1902) : Intraindividuelle Variabilitat der Carapax-dimen-
r -"""Sionen bei brachyuren Crustaceen ; Arch. Entw. Mech., 13, 588.
(191 7) : Wachstumsmessungen an Sphodromantis bioculata
Burm. III. Lange regenerierender und normaler Schreitbeine ;
Arch. Entw. Mech., 43, 1.
(1925) : Direkte Temperaturabhangigkeit der Schwanzlange bei
Ratten, Mus (Epimys) decumanus Pall, und M. (E.) rattus L. ;
Arch. Entw. Mech., 104, 434.
(1927) : Diskontinuitat des Wachstums als eine Ursache dis-
kontinuierlicher Variation bei Forficula ; Arch. Entw. Mech.,
112, 142.
(1930) : Connecting Laws in Animal Morphology. London :
Univ. London Press.
Purdy, C. and Sheard, C. (1931) : Electric Potentials in the Skin ;
Anat. Rec, 51, 25.
Raw, Frank (1927) : The Ontogenies of Trilobites, and their Signifi-
cance ; Anier. Journ. Sci., 14, 1-35, 131-49.
Ritchie, J. (1920) : The Influence of Man on Animal Life in Scot-
land ; Cambridge Univ. Press.
Robb, R. Cumming (1929) : On the Nature of Hereditary Size-limita-
tion. II. The growth of parts in relation to the whole ; Brit.
Journ. Exper. Biol., 6, 311-24.
Robbins, W. J.. Brody, S., Hogan, A. G., Jackson, C. M. and Greene,
W. (1928) : Growth. New Haven : Yale Univ. Press, 1928.
Rorig, (1901) : (Deer Antlers) ; Arch. Entw. Mech., 11, 65.
Ruud, Gudrun (1929) : Heteronom-orthotopische Transplantationen
von Extremitatenanlagen bei Axolotlembryonen ; Arch. Entw.
Mech., 118, 308-51.
Ruzicka, V. (1921) : Uber Protoplasmahysteresis und eine Methode
zur direkten Bestimmung derselben ; Pfl. Arch., 194, 135.
Santos, F. V. (1929) : Studies on Transplantation in Planaria ; Biol.
Bull., 57, 188-97.
Sasaki, K. (1928) : On the Growth of Telmessus cheiragonus ; Sci.
Rep. Tohoku Imp. Univ. (4th Ser., Biol.), 3, 805.
Scammon, R. E. and Calkins, L. A. (1929) : The Development and
Growth of the External Dimensions of the Human Body in
the Fetal Period. Univ. Minnesota Press : Minneapolis, p. 367.
254 PROBLEMS OF RELATIVE GROWTH
Schmalhausen, J. (1927A) : Beitrage zur quantitativen Analyse der
Formbildung. I. t)ber die Gesetzmassigkeiten des Embryon-
alen Wachstums : Arch. Entw. Mech., 109, 455-512.
(1927B) : Beitrage zur quantitativen Analyse der Formbildung.
II. Das Problem des proportionalen Wachstums ; Arch. Entw.
Mech., 110, 33-62.
(1930) : Ober Wachstumformeln und Wachstumstheorien ; Biol.
Zentralbl., 50, 292-307.
Schmalhausen, I. and Stepanova, J. (1926) : Das embryonale Wachs-
tum des Extremitattenskelettes des Huhnchens ; Arch. Entw.
Mech., 108, 729.
Schultz, A. H. (1926) : Fetal Growth of Man and other Primates ;
Quart. Rev. Biol., 1, 465.
(1930); The Skeleton of the Trunk and Limbs of Higher Primates ;
Human Biol., 2, 303.
Schwind, J. L. (1931) : Heteroplastic Experiments on the Limb
and Shoulder Girdle of Amblystoma ; Journ. Exper. Zool., 59,
265.
Severinghaus, A. E. (1930) : Gill Development in Amblystoma punc-
tatum ; Journ. Exper. Zool., 56, 1.
Sexton, E. W. (1924) : The Moulting and Growth-stages of Gammarus,
with descriptions of the Normals and Intersexes of G. chevreuxi ;
Journ. Mar. Biol. Assn., 13, 340.
Seymour Sewell, R. B. (1929) : The Copepoda of Indian Seas :
Calanoida ; Mem. Indian Mus., 10, 1-221.
Shaw, M. (1928) : A Contribution to the Study of Relative Growth
of parts in Inachus dorsettensis ; Brit. Journ. Exper. Biol., 6, 145.
Silvester, (1926) : Descrizione di particolori individui, etc. ; Boll.
Lab. Zool. Gen. (S- Agr., 19, 1.
Sinnott, E. W. (1930) : The Morphogenetic Relationships between
Cell and Organ in the Petiole of Acer ; Bull. Torrey Bot. Club,
57, 1-20.
Sinnott, E. W. and Durham, G. B. (1929) : Developmental History
of the Fruit in lines of Cucurbita pepo differing in fruit shape ;
Bot. Gazette, 87, 411-21.
Sinnott, E. W. and Hammond, D. (1930) : Factorial Balance in the
Determination of Fruit Shape in Cucurbita ; Amer. Nat., 64,
509-24.
Sjostedt, Y. (1925) : Revision der Termiten Afrikas ; K. Svensk.
Vetensk. akad. Handl. (3), 3, 1.
Smirnov, E. and Zhelochovtsev, A. N. (1926) : Veranderung der
Merkmale bei Calliphora erythrocephala Mg. unter dem Einfluss
verkurzter Ernahrungsperiode der Larve ; Arch. Entw. Mech.,
108, 579-95-
Smirnov, E. and Zhelochovtsev, A. N. (1931) : Das Gesetz der
Altersveranderungen der Blattform bei Tropaeolum majus L.
unter verschiedenen Beleuchtungsbedingungen : Planta Arch,
f. wissens. Bot., 15, 299-354.
Smith, G. W. (1906A) : Fauna and Flora des Golfes v. Neapel, Vol. 29
(Rhizocephala) .
(1906B) : High and Low Dimorphism ; Mit. Zool. Stat. Neapel,
17, 312.
BIBLIOGRAPHY 255
Snell, G. D. (1931) : Inheritance in the House Mouse: the linkage
relations of short-ear, hairless, and naked ; Genetics, 16, 42-74.
Sporn, E. (1926) : Uber die Gesetzmassigkeiten im Bau der Muschel-
gehause ; Arch. Entw. Mech., 108.
(1926) : Einiges iiber Ideal-, Normal- und Individual-formen
der Blatter ; Arch. Entw. Mech., 107.
Stockard, C. R. (1931) : The Physical Basis of Personality. New
York and London.
Streeter, G. L. (1930) : Deficiencies in Foetal Tissues. . . . Con-
tributions to Embryology, No. 126 (Carnegie Institution), 32.
Sturtevant, A. H. (1924) : An Interpretation of Orthogenesis ; Science,
59, 579-
Taylor, W. P. (1915) : Description of a new subgenus (Arborimus)
of Phenacomys, with a contribution to knowledge of the habits
and distribution of Phenacomys longicaudus True ; Proc. Cal.
Acad. Sci., 4th series, 5, 111-61.
Tazelaar, M. A. (1930) : The Relative Growth of parts in Palaemon
carcinus ; Brit. Journ. Exper. Biol., 7, 165.
Teissier, G. (1927) : La croissance nucleaire en fonction de la crois-
sance cellulaire au cours de l'ovogenese, chez Hydractinia
echinata (Flem.) ; C.R. Soc. Biol., 97, 1524.
(1929) : Dysharmonies chimiques dans la croissance larvaire de
Tenebrio molitor ; C.R. Soc. Biol., 100, 1171.
- (1931) : Recherches morphologiques et physiologiques sur la
croissance des insectes ; Trav. Stat. Biol. Roscoff, 9, 29—238.
Thiel, M. E. (1926) : Formwachstumversuche an Sphaerium corneum ;
Arch. Entw. Mech., 108, 87-137.
Thompson, C. B. (191 7) ; Origin of the Castes of the Common Termite,
Leucotermis flavipes. Journ. Morph., 31, 83.
Thompson, D'Arcy W. (1917) : Growth and Form : Cambridge Univ.
Press, p. 793.
Thomson, G. M. (1922) : The Naturalization of Plants and Animals in
New Zealand : Cambridge Univ. Press.
Todd, T. Wing ate (1926) : Skeletal Adjustment in Jaw Growth ;
Dental Cosmos, Dec, 1926, p. 17.
Tucker, B. W. (1930) : On the Effects of an Epicaridan parasite,
Gyge branchialis, on Upogebia littoralis ; Quart. Journ. Micros.
Sci., 74, 1-118.
Twitty, V. C. (1930) : Regulation in the Growth of Transplanted
Eyes ; Journ. Exper. Zool., 55, 43.
Twitty, V. C. and Schwind, J. L. (1931) : The Growth of Eyes and
Limbs transplanted heteroplastically between two species of
Amblystoma ; Journ. Exper. Zool., 59, 61-86.
Ubisch, L. v. (191 5) : Uber den Einfluss von Gleichgewichtsstorung
auf die Regenerationsgeschwindigkeit ; Arch. Entw. Mech., 41,
237-50.
Ubisch, L. von (1922) : Uber die Aktivierung regenerativer Potenzen ;
Arch. Entw. Mech., 51, 33-58.
(1923) : Das Differenzierungsgefalle des Amphibienkorpers und
seine Auswirkungen ; Arch. Entw. Mech., 52, 624-70.
256 PROBLEMS OF RELATIVE GROWTH
Vandel, A. (1930) : La production d'intercastes chez la fourmi Pheidole
pallidula sous Taction de parasites du genre Mermis ; Bull. Biol.,
64, 457-94-
Wachs, H. (1914) : Neue Versuche zur Wolff 'schen Linsenregeneration ;
Arch. Entw. Mech., 39.
Waddington, C. H. (1929) : Notes on graphical methods of recording
the dimensions of Ammonites ; Geol. Mag., 66, 180-6.
Wallis, R. Sawtell (1932) : Relative Growth of the Extremities
from Two to Eighteen Years of Age ; Amer. Journ. Phys.
Anlhrop. (cited from Wistar Inst. Advance Abstract-sheet
No. 179).
Wardlaw, C. W. (1924-1928) : Size in Relation to Internal Morphology.
1, 2 and 3 ; Trans. Roy. Soc. Edinb., 53 (1924), 54 (1925) and
56 (1928).
Watanabe, Y. (1931) : On the Physiological Axial Gradients of
Chaetopod Annelids. II. Axial Gradients of Oxidizable Sub-
stance in Earthworms as determined by the Manoilov Reaction.
Sci. Repts. Tohoku Imp. Univ. 4th Series (Biol.), 6, 437-73.
Weiss, P. (1926) : Morphodynamik : ein Einblick in die Gesetze der
organischen Gestaltung an Hand von experimentellen Ergeb-
nissen ; Abhandl. theoret. Biol., Heft 23 ; pp. 43.
Werner, F. (1927) : Wachstum und Formentwicklung der Cladocere
macrothrix rosea; Arch. Entw. Mech., 109, 241-52.
Wheeler, W. M. (1920) : Ants. New York : Columbia Univ. Press.
(1928) : Mermis Parasites and Intercastes among Ants ; Journ.
Exper. Zool., 50, 165.
Wilder, I. W. (1924) : The Relation of Growth to Metamorphosis
in Eurycea bislineata (Green); Journ. Exper. Zool., 40, 1-112.
Yerkes, R. M. (1901) : A Study of Variation in the Fiddler-crab
Gelasimus ; Proc. Amer. Acad. Arts. Sci., 36, 417.
Zeleny, C. (1905) : Compensatory Regulation ; Journ. Exper. Zool.,
2, 1.
Zuckerman, S. (1926) : Growth-changes in the Skull of the Baboon,
Papio porcarius ; Proc. Zool. Soc. (1926), 843.
ADDENDA
HERE I have summarized a few papers which I came
across too late to insert in the body of the book.
Owing to an oversight, the valuable work of Robb
(1929) has not been adequately discussed in the text. It has
very interesting bearings on the relation of heterogon to endo-
crine control (see Chap. VI, §4). He investigated the growth
of various organs in a large (Flemish) and small (Polish)
breed of rabbits, reaching about 6 and 3 kg. adult weight
respectively, and in their Fx hybrids.
He first found that the pituitary (weight) shows simple
negative heterogony relative to (clean) body- weight, with the
same growth-coefficient (k = 0-55) in all three types ; the
curves all have the same point of origin. Adrenal weight
shows simple positive heterogony, but k is higher for the small
breed (1-34 as against 1-19 for the Flemish). As a result the
relative weight attained by the adrenal in the adult Polish
is just double what it is in the adult Flemish (0-2 as against
o-i per cent) ; the hybrids show an intermediate value. The
difference is due almost exclusively to an enlargement of the
cortex.
The growth of the thyroid falls into two phases, one of
negative heterogony (k about 0-53) up to about 600 g. body-
weight, and one of positive heterogony (k about 1-12) from
then on.
The testis, like the adrenal, attains a higher relative weight
in the dwarf than in the giant race (0-3 as against 0-12 per
cent, with a value of 0-2 in the hybrids), and the heterogony
curves against body- weight are, of course, very dissimilar in
the two breeds. But when testis weight is plotted logarith-
mically against adrenal weight, the two breeds show almost
identical curves, negatively heterogonic (k = 0-74) up to 40
days of age, then with very high positive heterogony (k = 2-3).
Testis weight against pituitary weight, on the other hand,
shows k = 1-4 up to 40 days, but then k = 5-1 for the giant,
5-8 for the dwarf race. This appears to indicate a more
marked interdependence of testis and adrenal cortex than of
*7 257
258 PROBLEMS OF RELATIVE GROWTH
testis and pituitary. The identical growth-coefficient of the
pituitary in both races indicates that body-size here must
primarily depend upon other factors than relative pituitary size.
Robb concludes with an interesting theoretical discussion,
which cannot be summarized here. One point deserves
mention : if an organ has a certain fraction in active hetero-
genic relation with the body, but also an inert fraction which
is isogonic, then the formula for its growth will be y = bx -+- c.
In some cases apparently irregular heterogony curves could
be made to conform to the simple heterogony formula by
correcting for such an inert fraction ; but this for the moment
remains speculative.
Werner (1927) gives an elaborate discussion of form-changes
in the Cladoceran Macrothrix rosea. Certain parts appear to
show positive, others negative heterogony. But the growth-
changes are often complex, and would appear to indicate the
existence of elaborate gradient-fields controlling growth.
Anderson (193 1) finds that the simple heterogony formula
applies to the growth of various parts in the Cladoceran
Daphnia magna. Interestingly enough, while carapace length
is positively heterogonic (relative to total length) until the
time of maturity, after which it becomes isogonic or slightly
negatively heterogonic, carapace height shows positive heter-
ogony throughout life, although its growth-coefficient is lower
after maturity.
Adolph (1930) has an interesting note on the interrelated
effects of size and age upon metamorphosis in unoperated
frog larvae. He finds that for a given brood, (W — d) (A — e)
is a constant, where W is body- weight at completion of meta-
morphosis, A the age in days at which forelimbs appeared,
and d and e are constants. Thus no increase of size would
permit metamorphosis to occur before e days, but meta-
morphosis would never occur if the animal never attained to
body-weight d. We may conclude that the growth of the
thyroid (or of the thyroid-controlling agency of the pituitary)
normally shows a heterogonic relation to absolute size ; but
that time also, within limits, promotes its growth (see p. 39, n.).
Further evidence of change of proportions with change of
size in termites (see p. 65) is afforded by the work of Light
(1927). In this paper he confined himself to the soldier caste
of Coprotermes. Thirteen species of different absolute size
were measured. There was no correlation between absolute
size and certain characters such as the length-width ratio of
the head. But there was for other characters, e.g. the ratio
ADDENDA 259
of minimum to maximum breadth of head and of gula. The
minimum head-breadth is anterior; the maximum gular-
breadth is at a spot close to the maximum head-breadth, with
maximum gular-breadth more anterior. Accordingly we find
these ratios move in opposite directions with increasing
absolute size, that for head increasing steadily from below
o-6 to over 0-65, that for gula decreasing steadily from over
07 almost to 0-5. This means that with increased absolute
size, the lateral growth of the head (and its parts) is relatively
greater anteriorly than in the region of maximum width.
And this, we may presume, is correlated with a relative
increase of jaw-size, though Light gives no data on this point.
With reference to the conclusions of Hecht (p. 38), the
work of Keys (1928) also indicates that Hecht's assumption
of form-constancy in teleost fish is not strictly true. He finds
that in herrings, sardines, and Fundulus the weight increases
faster than the cube of the length, which implies form-change.
Somewhat similar results have been obtained by Hickling
(1930) for the dogfish Acanthias vulgaris.1
In connexion with the law of antero-posterior development,
and the graded changes it may induce (see p. 132), the follow-
ing point, which has been brought to my notice by Professor
R. J. S. McDowall, is of interest. Bray (1931) finds that the
incidence of eczema on different regions of the body varies
with age. The incidence for head and neck declines with
age, that for extremities increases with age, while that for
the trunk remains approximately constant. There is reason
to believe that the incidence of ringworm behaves in a some-
what similar way.
Gajewski (1922) discusses the effect of salinity upon Artemia
salina, whose form-changes continue long after sexual maturity.
Increasing salinity diminishes final absolute size, and also
changes bodily proportions, as seen in the following table (for
females : males are similar).
Salinity (Baumez). . 40 f 120 180 2 2° Be
Body-length, mm.. . 15-56 I3'°5 IO'65 9-62 7-8
Post-abdomen
— — ; ratio . o-Q2 1-02 1-20 1-30 1-42
abdomen
Length-breadth ratios :
Of 8th post-abdominal
segment . .2-3 27 3-5 4-5 5-0
Of 7th ditto . . 1-3 1-35 1-4 1-5 i-6
Of gill-sacs of 6th foot . 2-0 1-9 i-8 1-7
1 The same is indicated for the Bittering, Paracheilognathus, by the
data of Shaw (1931), Bull. Fan. Memor. Inst. Biol., 2, 245.
17*
260 PROBLEMS OF RELATIVE GROWTH
The furca diminishes disproportionately with increasing
salinity. It would be of great interest to investigate the
growth-coefficients of various parts accurately in different
salinities. This is an important addition to Chap. VI, § 7.
Murr (1929) x gives particulars concerning the relation
between retinal cells and the body as a whole in various
mammals, both as regards relative size, and relative develop-
mental rate. The results are interesting both as regards
individual ontogeny and comparative physiology.
Unpublished work which E. B. Ford kindly allows me to
quote, on relative eye-size in males of the red no-white mutant
of Gammarus chevreuxi at 230 C, shows that after a head-
length of a little over 0-5 mm. has been attained, up to the
maximum size (head-length 1-64 mm.) there is simple negative
heterogony both of eye-length (dorso-ventral) and eye-breadth
(antero-posterior) against head-length (antero-posterior), the
growth-coefficients (k) being 0-89 and 071 respectively.
Immediately after extrusion, however (head-length 0-25 — 0-3
mm.), both dimensions show positive heterogony (k nearly
2 -5 for eye-length and over 1-5 for eye-breadth), subsequently
diminishing regularly to reach their definitive values. As
consequence eye-breadth relative to eye-length shows a steady
negative heterogony with k a little below o-8 during the
' definite ' period and for a little before it, with a still lower
k value (o-6 to 0-65) for the earliest stages. The early phase
of positive eye-heterogony would seem to be due to the late
development of the organ, which is not fully differentiated
at the time of extrusion. (See Chap. IV, § 7.)
With reference to the correlation between growth-rate
and sensitivity to female hormone in fowl feathers (p. 101),
Lillie and Juhn (1931) find that this holds also in different
parts of the single feather. This growth-gradient has im-
portant consequences for the development of certain feather-
patterns.
With reference to bio-electric phenomena (p. 174), Purdy
and Sheard (1931) find in human beings that there is a definite
association of ' low metabolism with large differences of
electrical potential as measured at the extremities of the
body ' and vice versa.
J. W. Buchanan (1930, J. Exper. Zool., 57, 307 and 455)
establishes the existence of an antero-posterior osmotic
1 Murr, E. (1929), Zur Entwicklungsphysiologie des Auges, II,
Biol. Zentralbl., 49, 346.
ADDENDA 261
gradient concerned with water-appropriation, analyses its
action, and discusses it in relation to other ' axial gradients '
of Planarians (see pp. 171-2').
Olmsted and Baumberger (1923) state that in the crabs
Hemigrapsus oregenensis, H. nudus, and Pachygrapsus cras-
sipes, carapace length increases in a linear relation with
carapace width. Unfortunately they do not give their actual
measurements, and their graphs, in which individual points
are plotted, might equally well indicate slight heterogony.
This is especially so with P. crassipes, where length seems to
show slight positive heterogony relative to width. In H.
nudus the points are too few for any conclusion, and in H.
oregonensis the relation appears to show if anything a slight
negative heterogony. There must be marked heterogony of
the male chela, especially in H. oregonensis, where large males
have chelae up to 30 per cent of total weight, while the value
for large females is never over 7 per cent. In Carcinus maenas,
Huxley and Richards (1931) find a slight heterogony of carapace
length.
Mr. G. H. Locket kindly allows me to cite his results (unpub-
lished) on the chelicerae of the spider Theridion lineatum. The
jaw undergoes heterogonic growth and takes on its definitive
appearance only at the last moult, at which sexual maturity
is attained ; prior to this it appears to be almost isogonic.
(In the genus Linyphia, there are signs of heterogony at the
penultimate moult.) The jaw (basal joint, paturon) of the
adult male is much elongated, whereas that of the female
remains more nearly similar to that of the juvenile stages.
The adult jaw has a prominent tooth on its inner surface,
and measurements can be made of the lengths proximal and
distal to this. Sternal area was taken as standard body
measurement, and the square root of this was used as a stan-
dard against which to plot linear jaw measurements. There
is a moderate size-range in adult females, a considerable one
in adult males (probably dependent mainly on differences in
total moult-number). It is clear that in the formation of
the male type (i) jaw-breadth and jaw-length are both posi-
tively heterogonic, but length much more so than breadth
(k = about 1-9 as against about 1-3) ; (ii) in regard to length-
growth, the basal region, proximal to the tooth, is roughly
isogonic, while the distal region beyond the tooth is very
highly heterogonic (k = 27) ; (iii) the length of the falx
(distal joint, unguis) is highly heterogonic (k = 2-6), but less
262 PROBLEMS OF RELATIVE GROWTH
so than the distal region of the main jaw. Here we find
differential growth operative definitely within a single segment
of an appendage (compare pp. 81, 98).
In the female, the figures are more irregular, but jaw-area
and falx-length clearly show negative heterogony, while the
distal region of the jaw is distinctly positive. Jaw-length as
a whole is approximately isogonic, which means that jaw-
breadth must be negatively heterogonic.
In the spider Dolomedes plantarius, P. Bonnet (La Mue,
L'Autotomie et la Regeneration chez les Araignees, These,
Toulouse, 1930) in his Table 25 gives measurements of the
lengths of legs at all instars from 2nd to nth (adult) in two
individuals. From these the mean increases per moult can
be calculated ; and it is then found that all the legs are in-
creasing at the same rate — i.e. for this region of the body
there exists no growth-gradient comparable to that seen in
hermit-crabs, etc. (p. in).
W. Beebe (Tropical Wild Life in British Guiana, 1917,
chaps. 18 and 19) gives some data as to the heterogony of
the claws of the Jacana. That of the hind toe is greatest.
In the embryo the claws are not dissimilar to those of other
birds and the hind claw-length is about ^ of the toe-length ;
in half-grown chicks it is \, in adults about f.
The other claws grow more slowly, reaching about § of the
length of the hind-claw ; their relative breadth-growth is greater
than that of the hind-claw. The heterogony of the claws begins
only in the late embryo ; that of the toes begins much earlier.
He also refers to the bill of the aberrant cuckoo-like Ani,
which is swollen in the adult. It, however does not begin
its positive heterogony until after the young bird has left the
nest. At hatching, it is typically cuckoo-like, though some-
what swollen. It would be interesting to obtain accurate
measurements on these structures. The case of the Jacana is
interesting, since the great length of toes and claws is quite defin-
itely adaptive, allowing the bird to walk over floating leaves.
G. Duncker (1903 ; Biometrika, 2, 307) attacks the problems
of growth-correlation and asymmetry in male fiddler-crabs
(see pp. 80 and 121) by means of standard biometrical
methods. These, however, failed to give any very important
biological information, e.g. as to growth-centres or growth-
gradients, or at least nothing so clear-cut as is to be obtained
by the simple methods of taking means for a number of size-
classes. The growth-gradient in the large chela is indicated
ADDENDA 263
by the fact that the correlation between right and left side
diminishes distally along the appendage. The correlations
for lengths of merus, carpus and propus are, for right-handed
males, 0754, 0-698 and 0-473 ; for left-handed males, 0-789,
0-699, °'549- He suggests that the chela asymmetry may be
responsible for the asymmetry of other parts.
Marples (Proc. Zool. Soc, 1931, p. 997) has given some
interesting facts as to the percentage changes of different
parts of birds' wings during development. For calculating
growth-coefficients, he has kindly put at my disposal his
original data on the Common Tern (the species on which
the most numerous measurements were taken). For a proper
analysis, considerably more measurements are needed, includ-
ing measurements of some standard part of the body ; but
provisionally we can see that there are three distinct phases
of growth, during each of which the relative growth-rates of
different parts of the wing remain approximately constant.
The first phase ends at hatching ; the second, juvenile phase
goes from hatching (wing-length below 40 mm.) to wing-
length about 100 mm. ; the third up to the largest adults
(wing-length over 180 mm.).
The growth-coefficients (approximate only) of the lengths
of ulna and radius, relative to humerus length, are as follows :
Phase. r 23
Ulna . . . . .1-05 about o-8 1-6
Radius . . . . .1-2 ,, o-8 1-45
The growth-gradient appears to centre in the radius in the
1st phase (though this may be due in part to the late differ-
entiation of this terminal region) ; to be reversed, centering
in the upper arm, in the 2nd phase ; and again to change its
form, centering in the fore-arm, in the final phase (cf. p. 34).
I have not discussed the enormous body of data given in
Donaldson s The Rat (1924), since all the comprehensive tables
there set forth do not give the actual values for the various
organs, but calculated values. These values have been
calculated in accordance with empirical formulae devised by
Hatai to fit smooth curves to the data.
These formulae are of the following types :
y = a log x + b
y = a (log x -f- c) -f- b
y = ax + b log x + c
y = ax -f b (log % -f- c) -\- d
y = (ax -f- b) -f- b (log x 4- c) + d
y = axb
264 PROBLEMS OF RELATIVE GROWTH
where y = organ-size, % = body-size, and a, b, c, d are con-
stants. Most of these have no assignable biological signi-
ficance.
If the measurements contained in the original papers were
re-analysed, it is probable that a number of cases of simple
heterogony would be revealed. I have done this for one or
two organs. E.g. testis weight (S. Hatai, 1913, Am. J. Anat.,
15, 8y), after an early period of rather slow growth, where
more data are needed, shows a good approximation to simple
heterogony between body- weights 25 g. and 95 g., with growth-
coefficient about 1-65. After this, it enters on a phase of
negative heterogony, with k only about 0-4 to 0-45. The
ovary shows a very similar set of three phases, but the points
are more irregular.
The hypophysis shows an interesting sex-difference. From
body-weights of 60 g. on, k for the male hypophysis is positive
(k about 1-3), whereas for the female it is negative (k rather
below o-8). Below this size, the points are rather irregular,
but those for both sexes appear to fall on a prolongation of
the curve for large females.
For heart -weight (males) k is close to o-8 from body- weight
140 g. on. Before that, the points are more scattered, but
could be considered as fitting the same curve, though appar-
ently with a temporary acceleration of relative growth from
body- weight 60 to 120 g., later compensated for.
The kidneys (males) begin by being somewhat negatively
heterogonic (k about 075), and then, after an irregular period,
show definite positive heterogony from body- weight 180 g. on,
with k close to i-i. Lung- weight (males) is more irregular,
but roughly approximates to a negative heterogony of growth-
coefficient about o-8 throughout. For further analyses along
these lines the data should be re-grouped into larger size-
classes.
T. C. Byerley (1932, J. Exp. Biol., 9, 15) has recently shown
that in chick embryos allantois weight shows negative hetero-
gony relative to egg- weight, being roughly proportional to the
two-thirds power of egg- weight. It may be recalled that egg-
weight, at least in large birds, is itself roughly proportional
to the two-thirds power of body- weight (p. 226).
He further points out an important connexion between rela-
tive and absolute growth. In fowls, gut-weight shows negative
heterogony (H. B. Latimer, 1924, /. Agr. Res., 29, 363).
Byerley finds a linear relation between feed-consumption and
ADDENDA 265
gut- weight (up to sexual maturity), Accordingly there will
be a steady decline in the amount of food ingested per unit
of body-weight ; and this may be presumed to be responsible
for the steady decline in percentage absolute growth-rate
during this period.
INDEX OF AUTHORS
Abe, Y., 201 n.
Abeloos, M., 4, 21, 53, 138, 172, 174,
179
Adolph, E. F., 39, 258
Aikman, J. M., 133
Allen, B. M., 181-2
Allen, J. A., 41 n.
Alpatov, W. W., 65
Alverdes, F., 173
Anderson, B. G., 258
Appleton, A. B., 39 n.
Arrow, Dr., 214
Atkins, D., 94
Baer, K. von, 237, 239
Baillie-Grohman, W. A., 45
Balfour-Browne, F., 17
Banks, E., 15, 23, 62
Bateson, W., 117
Bateson, W., and Brindley, H. H., 59,
72. 76
Baumberger, J. P., 261
Bean, R. B., 203
Becher, A., 14, 18
Beebe, W., 235, 262
Beer, G. R. de, 237, 239
Belehradek, J., and Huxley, J. S., 229
Benazzi, M., 85
Benoit, J., 177
Berkson, J., 203
Berrill, N. J., 172
Bertalanffy, L., 119, 171
Blyth, J. S. S., Dodds, E. C, and
Gallimore, E. J., 178
Boas, F., 221
Bolk, L., 239
Bonnet, P., 262
Boveri, Th., 112, 171
Bower, F. O., 223-4
Bray, G., 250
Brindley, H. H., 59, 72, 76
Buchanan, J. W., 172, 260
Bush, S. F., 17, 61, 62, 87, 91, in, 138
Bush, S. F., and Huxley, J. S., 91, 11 1
Callow, F. S., 85, 121
Caiman, W. T., 117, 137
Champy, C, 1, 42, 55, 57, 115 ;;., 179,
180, 182, 188, 205 n., 213-14, 216,
219 n.
Child, C. M., 112, 129, 170-6, 179
Choi, M. H., 197
Clausen, H. J., 173
Coghill, G. E., 203
Collip, J. B., 178
Cott, H. B., 17, 87
Crew, F. A. E., 200
Cushing, H., 188
Davenport, C. B., 203 n., 232
Davenport, C. B., and Swingle,
W. W., 131
Dean, I., 82, 85, 90
Diakonov, D. M., 72, 74
Dombrowski, E. von, 42, 45,
Domm, L. V., and Juhn, M., 227
Donaldson, H. M., 1, 16, 263
Dubois, E., 17, 138, 215, 224-5
Dudich, E., 59, 60, 76, 208-10
Durham, G. B., 234
Duncker, G., 262
Edmonds, E., 65
Edwards, A. S., 113
Edwards, A. S., and Huxley, j. S.,
113
Eigenbrodt, H. J., 63
Emerson, A. E., 65 «., 67
Emery, C, 63
Entz, G., 200
Faulkner, O. H., 100-2
Faure-Fremiet, E., 117
Ford, E., 116, 117
Ford, E. B., 147, 229, 260
Ford, E. B., and Huxley, J. S., 4,
229-231, 238-9, 242
Fulton, J. F., 214
Gabritchevsky, E., 125
Gajewski, N., 200, 259
Gause, C. F., 63
Giesbrecht, W., 118
Goldschmidt, R., 4, 119, 229, 234
Gray, J., n
Griffin, A., 208, 210
Grohman, W. A. Baillie-, 45
Gustavsen, R. G., 100-2
Guyenot, E., 117
Guyenot, E., and Ponsc, K., 171, 175
267
268
PROBLEMS OF RELATIVE GROWTH
Haeckel, E., 237, 239
Haldane, J. B. S., 81
Hammett, F. S., 183-8
Hammond, D.( 234
Hammond, J., 17, 88-90, 138, 201-2,
222
Hanby, A. M., 202
Hardesty, M., 178
Hare, F., 66, 67
Harris, H. A., 203
Harrison, R. G., 7 n., 191-6
Hasebrock, K., 224
Haseman, J. D., 168 n.
Hatai, S., 16, 263-4
Heath, A., 65, 67
Hecht, S., 38, 100, 259
Henderson, J. R., and Mathai, G., 17,
85
Herms, W. B., 63
Herrick, F. G., 118
Hersh, A. H., 15, 21, 234
Hesse, R., 222
Hickling, C. F., 259
Hinton, M. A. C., 40
Hirsch, G. C, 171
Hooton, E. A., 220-1
Huntsman, A. G., 200
Hutt, F. B., 182
Huxley, J. S., 4, 8, 9, 14, 17, 18, 20,
34, 42, 45, 59, 61, 62, 72, 76, 82, 83,
87, 88, 96, 113, 116, 119, 123, 150,
166, 197, 205-8, 210, 218, 225, 229-
31, 238-9, 242.
Huxley, J. S., and Callow, F. S., 85,
121
Huxley, J. S., and Fulton, J. F., 214
Huxley, J. S.,and Richards, O. W., 20,
261
Imms, A. D., 67
Jackson, C. M., 16, 39 n., 197, 201
John, 67
Jucci, C, 78
Juhn, M., 227, 260
Juhn, M., Faulkner, O. H., and Gus-
tavsen, R. G., 100-2
Kalshoven, L. G., 66
Kearney, 38 n.
Keith, A., 130, 188
Kemp, S., 17, 33, 82, 85, 90
Keys, A. B., 38, 259
Kieslinger, A., 239
Klatt, B., 17, 208, 215, 224
Kozelka, A. W., 178
Krizenecki, J., 174
Kuhl, W., 72 n.
Kunkel, B. W., and Robertson, J. A.,
34, 40, 138
Lameere, A., 213, 214
Landauer, W., 131 n.
Lapicque, L., 17, 138, 215, 225
Lapicque, L., and Giroud, A., 17
Latimer, H. B., and Aikman, J. M.,
133
Lebour, M., 159
Levy, H., 6, 157
Light, S. F., 258
Lillie, F. R., and Juhn, M., 260
Lipschiitz., A., 177
Locket, G. H., 261
Lund, E. J., 174
McDowall, R. J. S., 259
Mackintosh, N. A., and Wheeler,
J. F. G., 16, 135-7
Marples, B. J., 263
Martin, R., 1
Mathai, G., 17, 85
Matthew, W. D., no
Mead, C. H., 129
Miller, W. F. C, 102
Mjoberg, E., 237
Morgan, A. H., 237
Morgan, T. H., 32, 167
Murr, E., 224, 260
Nahagas, J. C, 128
Needham, J., 119, 171
Nevalonnyi, M., and Podhradsky, J.,
187
Nomura, E., 15
Olmsted, J. M. D., and Baumberger,
J. P., 261
Osborn, H. F., 218
Parkes, A. S., 179
Parrot, C, 225
Parsons, F. G., 219-20
Pearsall, W. H., 14, 16, 21
Pearsall, W. H., and Hanby, A. M.,
202
Pearson, J., 93
Perkins, M., 169, 197-9
Petersen, Chr., 161 n.
Pezard, A., 8, 177-8
Podhradsky, J., 187
Ponse, K., 171, 175
Przibram, H., 1, 7 »., 17, 40, 51 n., 53,
68, 72, 96, 125-7, !35. 165, 174,
175 n., 189-191, 200, 203
Purdy, C, and Sheard, C, 174, 260
INDEX OF AUTHORS
269
Ratcliffe, F. N., 169
Raw, F., 134
Richards, O. W., 20, 261
Ritchie, J., 205-7
Robb, R. C, 49 n., 176 n., 227, 257
Robbins, W. J., Brody, S., et. al., 38 n.
Robertson, J. A., 34, 40, 138
Rorig, 42
Ruud, G., 135
Ruzicka, V., 7
Santos, F. V., 172
Sasaki, K., 14, 15, 93
Scammon, R. E., 132-3, 136-7. 241
Scammon, R. E., and Calkins, L. A.,
132
Schmalhausen, I., 7, 139-47
Schmalhausen, I., and Stepanova, J.,
118
Schultz, A. H., 137, 234
Schwind, J. L., 49M., 191. i94M->
195 n.
Seymour Sewell, R. B., 85-6, 123, 133
Sexton, E. W., 40
Shaw, M. E., 14, 34, 35, 55 »■. 68, 123
Shaw, T. H., 259 n.
Sheard, C, 174, 260
Silvester, 66 n.
Sinnott, E. W., 174
Sinnott, E. W., and Durham, G. B.,
234 _
Sinnott, E. W., and Hammond, D.
Sjostedt, Y., 66
Smirnov, E., and Zhelochovstev,
A. N., 63, 65, 163, 176 n.
Smith, G. W., 34, 35, 53-5, 70, 199,
213
Snell, G. D., 234
Spemann, H., 172
Sporn, E., 163
Stepanova, J., 118
Stockard, C. R., 129-31, 172
Sturtevant, A. H., 218
Sumner, F. B., 40
Suster, I., 175 ».
Swingle, W. W., 131
Taylor, W. P., 15, 22
Tazelaar, M. A., 14, 34, 91, 121
Teissier, G., 16, 17, 21, 25, 58, 61 n.,
231
Thiel, M. E.( 199, 203
Thompson, C. B., 67
Thompson, D'Arcy W., 1, 3, 79, 87,
104-10, 150-4, 157-61, I7X» 222,
244
Thomson, G. M., 207
Todd, T. W., 95
Tucker, B. W., 14, 98, 99
Twitty, V. C, 51, 197
Twitty, V. C, and Schwind, J. L.,
49 n., 191, 194 n.
Ubisch, L. von, 112, 126, 135, 171,
173
Vandel, A., 66 n., 199 n.
Wachs, H., 51
Waddington, C. H., 158 n.
Wallis, R. S.,233
Wardlaw, C. W., 223
Watanabe, Y., 174
Weiss, P., 112, 171
Werner, F., 176 n., 258
Wheeler, S. F. G., 16, 135-7
Wheeler, W. M.( 61, 66 n., 199 n.
Wilder, I. W., 17, 203
Yerkes, R. M., 121
Zeleny, C, 166, 189, 190
Zhelochovstev, A. N., 63, 65, 163,
176 n.
Zuckerman, S., 14, 18, 19
SUBJECT INDEX
{Figures in italics refer to pages with illustrations of the subject cited.)
abdomen, crustacean, 14, 75, 16, 17,
20, 34, 35, 68, 69, 86m., g3, 94,
95. 239
achondroplasia, 131
adaptation, 214, 219
adrenal, 257-8
adult phase, 39
age, physiological, 175
anencephaly, man, 127, 130
angle, constant, of logarithmic spiral,
157
antennae, 55, in, 112, 113, 114, 115,
175
— copepods, 85, 86
antero-posterior differentiation ,
ii8m., 132 seq., 139
antlers, deer, 17, 42 seq., 43, 44, 46,
4J, 48, 205, 206, 207, 210, 216,
218-19, 227
ash content, mealworm, 29
asymmetry, fiddler-crabs, 121, 166,
189
— hermit-crabs, 112, 190
— in heterochely, 96 seq., 189-91
auxano-differentiation, 120, 148, 232
axial gradients, 139, 171-6, 242
beak, Hornbills, 15, 23
beard, man, 102
bimodality, yi, 72 n., 73
bio-electric potential, 174, 260
body-build, man, 232, 233
brain, chick, 145, 146
— mammals, 17, 215-16, 225
— rat, 164, 185
breadth-growth, and depth-growth,
96, gy, 100, 200
— and length-growth, 95, 96, 97, 96',
100, 114, 115, 173, 200, 203, 234
carapace, crustacean, 23, 24, 25, 27,
106, loy, 203, 258, 261
Cartesian co-ordinates, 104, 105, 106,
J07, 108, iog
castration, 177-8, 183
chelae, 8 seq., 14, ly, 31, 33, 34, 38, 53,
55. 69, 70, 80, 82, 83, 84, 85, 90,
gi, g2, 96, gy, g8, 99, 119, 121,
166, 167-9, 216, 22y, 238, 262-3
chelicerae, spiders, 261
chemo-differentiation, 119, 150, 168,
169
comb, fowls, 177-8
combustion, heat of, 28, 29, 30
correlation, vii, 221
— of growth-effects, 120 seq.
cranium, mammals, 14, 18, ig
crest, newts, 179, 180
crusher claw, 96, gy, 189-9J
cyclopia, 172
Developmental Direction, law of, 132
seq.
differentiation, direction of, 168 n.
— law of antero-posterior, 118 n.,
132 seq., 139
digits, proportions of, 130, 188, 235,
262
dimorphism, 68 seq.
— developmental, 70
— environmental, 71
— genetic, 70
diphasic species, 70
dominant region, 172, 173, 175
dwarfs, ateliotic, 131
— mongoloid, 131
eczema, 259
eggs, relative size in birds, 225-26
elytron, beetles, 25, 114
environment, influence upon relative
growth, 197 seq.
equilibrium-position, 36, 232
evolution, 3, 106 seq., 216 seq.
eye, Drosophila, 14, 21, 22
— stick-insect, 27
eye-colour, Drosophila, 231-2
— Gammarus, 228, 229, 230-2, 237-
38
— man, 232
eyes, grafted, 51, ig2, ig3, ig4, [95,
196, 197
— rat, 184, 185
270
SUBJECT INDEX
271
fai 1 , mammals, 1 4
fat-content, mealworm, j<»
— wax-moth, 30
feathers, growth-rates of, 100-2, 264
field, morphogenetic, 117, 152, 171,
175, 176 n., 258
fish, proportions of, 37, 38, 259
forceps, of male earwigs, 71 seq.
form-changes, 39-41
function, effect of , 115, 119, 222, 224,
227
ganglia, 29
— asymmetrical, in Uca, 16S
genetics, 4
— and proportions of parts, 204, 207
gigantism, 131
gizzard, birds, 146
glutathione, 169
gnathopod, 36, 38
gradient, osmotic, 260-1
gradients, axial, 139, 171-6, 242
— in bio-electric potential, 174, 260
■ — in chemical substances, 169, iy<>,
174
- in differentiation, 174
in variability, 174
— metabolic, 171 n.
qualitative differences between,
176
-temperature, 172
grafting, 51, 52, 135
178, 191-7
growth, absolute, 6
— accretionary, 149 seq.
— additive, 149
- by cell-multiplication and cell size
increase, 184-5
— differential, 1
— equilibrium of, 50, 55
— in open and closed systems, 60
— in various planes of space, 95-100
— limited, 38 n., 39, 40
— linear, 132, 137, 140
— multiplicative, 11, 149
— relative, in embryonic life
— relative, limitation of, 49
— two phases of, 118 seq.
— unlimited, 38 n., 39, 40
growth-centres, 83, 85, 91-5
growth-coefficient, 8, 58, 81,
— corrected, 146
growth-constants, 145
growth-gradients, 3, jgseq., 105, 159,
160, 174-6
— determination of, 167 seq.
— ■ distribution in body, 104 seq.
— form of, 90 seq.
— general, ill seq., 152
^l, 167, [73,
139 seq.
60
186
growth-gradients in epidermal
structures, 100-2
— major and minor, no, 112
— positive and negative, 87 seq.
— steepness of, 83 seq., 114, 131, 172
growth-intensity, 3, 139
— local, effect of, 122, 123, 124, 125
— specific, 91 seq., 227
growth-partition, 3, 49, 51, 57, 50,
165
growth-potential, 80, 81, 97
— general distribution in body, 104
seq.
growth-profile, 116
growth-quotients, 135-41
growth-rate, relative, 6, 9
growth-ratio, lateral, in Mollusc
shells, 159
— length-width, in Mollusc shells,
154
— median, in Mollusc shells, 155
growth-ratios, constant, 4, 7, 8, n,
30, 38, 51. 57. *54 seq., 165,
234
examples of, 13 seq.
Haeckel's Law, 237 seq.
head, whales, 16, 136, 137
heart, birds, 224-5
— chick, 143
— rat, 184, 185
— vertebrates, 17, 264
heterochely, 96 seq., 189 seq., igi
heterogony, 5, 8
— and secondary sex characters,
188-9
— and sexual maturity, 8
— in related species, 32
— onset of, 31
— positive and negative, 17, 87 seq.,
"3
heterogony-mechanism, and detailed
form-changes, 217
heteromorphosis, 172, 175
' high ' males, 55, 70-6
histo-differentiation, 120, 232
histolysis, 173
hormones, and heterogony, 176 seq.,
257
— and specificity, 177, 181, 188
— sensitivity to, 101, 177, 181, 186,
260
horns, beetles, 17, 55, 115 n., 188, 213,
217
— Peridinians, 200
— vertebrates, 150-4, 173, 188, 218-
19
hypertrophy, functional, 115, 222,
224, 227
272
PROBLEMS OF RELATIVE GROWTH
insects, holometabolous, 55 seq., 115,
231
— neuter, 61 seq.
insulin, 214
interaction, of organs of different
growth-intensities, 195, 196
intersexuality, 36
isogony, 8, 37
kidney, chick, 143, 145
— rat, 184, 264
Lameere, phenomenon of, 2x3-14
leaves, plants, 14, 16, 176 n., 202
lens, chick, 144
— interaction with growth of optic
cup, 195, J96
light, effect on growth, 200
limb-buds, Urodele, 135, 192
limbs, Amphibian, and thyroid, 179
seq., 188
— chick, 142, 143, 144, 146
— Crustacea, 137
— fowls, 183, 187
— insects, 17, 25, 55, 56, 113, 114,
126, 127, 128
— man, 138
— sheep, 17, 88 seq.
— spiders, 262
' low ' males, 55, 70-6
mandibles, beetles, 17, 25, 55, 58, 59,
113, 114, 209-12
mane, horse, 102
mass-factor, 146
mathematics, and biology, 2, 244
maxilliped, and chela-growth, 123
metabolic gradients, 171
metamorphosis, 120, 179-82
morphogenetic field, 117, 153, 171,
175, 176 n., 258
moulting, and dimorphism, 68 seq.
moults, additional, 73 seq.
multimodality, 73, 78
mutation, and growth-gradients, 222,
234
— and growth-ratios, 234
myxoedema, 131 n.
neoteny, 67, 237-40
nerve-fibres, mammals, 17
nervous system, and asymmetry of
Uca, 168, 169
neurons, mammalian, 17
New Zealand, introduction of Red
Deer, 207
nipper claw, 96, 97, 189-91
nitrogen content, mealworm, 26, 29
nucleus, neuron, 27
nucleus, oocyte, 17, 21
nutrition, and Forficula forceps, 74,
75
— and relative growth, 193-4, T97.
198, 199, 201, 202
ommatidia, crustacean, 24
— insect, 27
oocyte, Hydractinia, 17, 21
operculum, of Hydroides, 189
optic cup, interaction with growth of
lens, 195, 196
organizer, in development, 172
orthogenesis, 218-19
oxygen consumption, 29
ovary, 264
paedomorphosis, 239-40
parasites, effect on host's growth, 38,
66 m., 99, 197-9
partition-coefficient, 3, 49, 227
pelvis, avian, evolution of, 106, 108
pereiopods, and chela-growth, 121 seq.
— crabs, 17, 137
periodicity, of growth, 203
peristalsis, 173
phosphorus-content, mealworm, 28,
29
— wax-moth, 30
physiology, comparative, 3
pituitary, 178, 186, 187, 188, 227,
257-8, 264
placenta-extract, 178
polarity, 119
position, spatial and morphological,
116
precocial young, 87, 90
proportions of parts, change of, 38-41,
88, 106, 182-8, 200, 201-2, 216-
23, 224-9, 257, 259
pupa, insect, 60
rate-genes, 4, 228, 229, 230-40
recapitulation, 4, 234-40
regeneration, and growth, 50, 51 n.,
53, 85, 100-2, 125-8, 165-7
— and morphogenetic fields, 152
— rate of, 166, 190
regeneration-gradient, 85, 173
regression, of antlers, 44
— of fowl's comb, 1 78
— of newt's crest, 179, 180
— of starved Planarians, 175, 179
regulation, of size, 52, 191-7, 228-9
retinal cells, 260
rhythms, of growth, 203
ringworm, 259
root, plants, 13, 14, 21
SUBJECT INDEX
273
salinity, effect upon Artemia, 259
Sardinia, dwarf deer, 208
selection, and environment, 202
— and growth-gradients, 88, 222, 223
sensitivity to hormones, and growth-
rate, 10 1, 184-6, 260
sex-hormones, 177-9, 182, 188
shells, 145 seq.
— bivalve, 162
— conical, 155
— foraminiferan, 163
— logarithmic-spiral, 155 seq., 156
— turbinate-spiral, 159 seq., 160
shoot, plants, 13, 14, 21
shoulder-girdle, grafted, 195 n.
size, adaptive increase of, 218-19
— and differentiation, 39 n., 175, 179,
181-2, 207, 2og, 212, 213, 222-4,
258
— limited by heterogony, 32
— non-adaptive increase of, 220-1
— range of in Lucanidae, 76
skeleton, fowl, 187
— rat, 184-6
skull, evolution of, in horse, 107, iog
— form of, in man, 219-2J
soldiers, ants, 61 seq.
— termites, 65 seq., 258
spiral, logarithmic, 151 seq.
— turbinate, 159 seq.
starvation, 179, 180, 201, 202
sub-species, 205, 208, 2og-n
sulphur compounds, and growth, 169,
170
sulphydryl, 169, iyo
systematics, 3, 204-14
tail, man, 235
— mice, 15, 17, 22, 200
— salamander, 17
' tail ', Papilio, 55, 57, 188
taxonomy, 3, 204-14
teeth, 150
temperature, and relative growth, 200
temperature-gradients, 172
tendons, relative size of, 229
teratology, 127 seq.
testis, 177-8, 227, 257, 264
thyroid, 181, 183-8, 257
time of origin of organs, 140 seq.,
201-2
time-relations, growth and differen-
tiation, 133, 140 seq., 201-2, 232,
236-40
tracheidai tract, relative size, 223-4
trimodality, 73, 76
undernutrition, 201, 202
vestigial organs, 235, 236, 237
viscosity, and age, 7
von Baer's Law, 237 seq.
water content, mealworm, 26, 29
— — wax-moth, 29, 31
wing-rudiments, dragonflies, 17
wings, birds, 263
— insect, 17, 113
workers, ants, 61 seq.
— termites, 66 seq.
work-growth ratio, 186
INDEX OF ORGANISMS
(Figures in italics refer to pages with illustrations of the organisms cited.)
Acanthias, 259
Acanthocinus, 55
Acanthotermes, 66
Acarines, 78
Aeschna, 85
Alpheus, 34, 96, 189
Amblystoma, 50, 52, 191, 1Q4, 796,
197, I98, 227
Ammonites, 155, 156, 15812., 161
Analges, 77, 78
Anguilla, 117
Ani, 262
Anomia, 163
Anomma, 61, 62, 64
Anoplocnemis, 55
Antelope, 153, 188
Antheracoceros, 23
Anthropoid ape, 225, 234
Ants, 61-4
Anura, 181
Apatornis, 106, 108
Apteryx, 226
Archaeopteryx, 106, 108
Armitermes, 67
Artemia, 259
Axolotl, 183 n.
Baboon, 14, iS, ig
Balaenoptera, 136
Basset-hound, 131
Beetles, 55, 56
Birgus, 216
Blatta, 27
Blowfly, 62, 65
Brachiopod, 149, 162
Brachyura, 216, 239
Brassica, 21
Calliphora, 65
Camponotus, 62, 64
Campsurus, 237
Cancer, 23, 27, 93, 169, 770
Capercaillie, 225
Capreolus, 45, 46, 47
Carcinus, 16, 20, 24, 25, 27, 34, 169
Cardium, 162
Carinatae, 226
Carrot, 13, 21
Cat, 133, 134, 225
Cephalopod, 155
Cervidae, 218
Cervus, 42, 43, 44, 45, 47, 48, 205
Chaffinch, 225
Chameleon, 188
Chaoborus, 27
Chick, 118, 140, 143
Cladoceran, 258
Clausilia, 161
Cloe, 126, 135, 173, 174
Clupea, harengus, 116
— pilchardus, 116
— ■ sprattus, 116
Cockle, 162
Cockroach, 27
Coleoptera, 214
Copepods, 86, 117, 133, 137
Coprotermes, 258
Cotton, 13, 21
Crab, 14, 23, 24, 25, 93, 107, 169, 189,
203, 239, 261
Crayfish, 27
Crustacea, 68
Cuculus, 227
Cucurbita, 234
Cyclommatus, 58, 59, 60, 76, 208, 209,
211
Daphnia, 258
Daucus, 21
Decapoda, 14
Deer, 42-8, 188, 205, 206, 208, 210,
227
Dentalium, 155
Diodon, 104, 105, 222
Dixippus, 27
Dog, 14, 18, 131, 215
Dogfish, 259
Dolomedes, 262
Donax, 162 n.
Dragonfly 17, 85,
Drosophila, 15, 21, 22, 63, 231
Dynastidae, 55
Dytiscus, 25
274
INDEX OF ORGANISMS
275
Earthworm, 174
Earwig, 71-5
Eel, 117
Elasmotherium, 151
Embolotherium , 219 n.
Enema, 188
Equus, no
Eriphia, 23, 27, 189
Eucalanus, 86
Euchirus, 56
Eupagurus, gi, in, 112, 119
Eurycea, 17, 203
Falcon, peregrine, 225
Fiddler-crab, 8-12, 31-5, 41, 80, 168,
262
Fish, 37
Foraminifera, 163
Forficula, 71-5
Fowl, 100, 101, 131, 183, 187
Frog, 182, 214, 258
Fundulus, 172, 259
Galleria, 16
Gammarus, 34, 40, 228, 237, 238, 241,
260
Gastropod, 156, 157, 161
Gibbon, 234, 238
Giraffe, 88
Goat, 153
Goliathidae, 217
Golofa, 213
Gossypium, 21
Gourd, 234
Gyge, 98
Haliotis, 159
Helix, 161, 239
Hemigrapsus, 261
Hemiptera, 55
Hermit-crab, 17, gi, 111, 112
Herring, 116, 259
Hobby, 225
Homarus, 96, 97, 169, 170, 189
Hordeum, 21
Hornbill, 15, 23
Horse, 107
House-fly, 63
Hydractinia, 17, 21
Hydroid, 172, 189
Inachus, 36, 38, 53, 54, 6g, 122, 123,
124
Insects, 17, 55, 68
Jacana, 235, 262
Lamellibranch, 162
Limacina, 759
Linum, 21
Linyphia, 261
Lobster, 96, g7, 118, 120, 169, 189
Lucanidae, 55, 58, 59, 76, 113, 207,
208, 210
Lucanus, 25, 58, 59, 76, 78, 114, 115
Lycidae, 237
Macrothrix, 258
Magpie, 225
Maia, 34, 82, 83, 84, 122, 123
Mammals, 17
Man, 127-33, 203, 219-2J, 225, 232,
233, 235. 259
Mantids, 174
Mayfly, 237
Mealworm, 16, 25, 174
Megaceros, 216
Mesohippus, 107, iog
Microtinae, 40
Miohippus, 107, iog
Mite, 77, 78
Mollusca, 15, 149, 154, 160, 203
Monkey, 225
Mouflon, 88
Mouse, 15, 22, 200, 225, 234
Mussel, 200
Mytilus, 200
Nannocalanus, 86
Nasturtium, 14, 16, 176 n.
Nautilus, 156, 157
Neotoma, 41 11.
Newt, 179, 180
Notonecta, 25, 27
Ocypoda, 17, 87, 216
Orthagoriscus, 104, 222
Orthopristis, 37
Orthoptera, 17
Oryctes, 115
Ox, 88
Pachygrapsus, 25, 27, 261
Palaemon, 17, 33, 34, 82, 84, 85, 90,
91, 92, 123, 124, 125, 175
Pandalus, 170
Papilio, 55, 57, 188
Papio, 18, ig
Parahippus, 107, iog, no
Pea, 13, 21
Pea-crab, 37, g4
Pecten, 161, 163
Peridinians, 200
Periplaneta, 170
Phanaeus, 188
Pheidole, 63
Phenacomys, 15, 22
Pilchard, no
276
PROBLEMS OF RELATIVE GROWTH
Pinnotheres, 37, 94, 95. "9
Pistol-crab, 34, 96
Pisum, 21
Planarians, 14, 21, 53, 172, 174, 179,
261
Planorbis, 156, 160
Plants, 14
Portunus, 166, 189
Potamobius, 27
Potamogeton, 202
Prawn, 34, 84, 92, 123, 125
Productus, 163
Protohippus, 107, 109
Psilotum, 223
Pteropod, 159, 163
Pupa, 161
Rabbit, 201, 202, 215, 227, 257
Rat, 1, 16, 183, 185, 187, 200, 201,
202, 263
Ratitae, 226
Razor-shell, 162
Red-breast, 215
Red-deer, 42, 43, 44, 47, 48
Reindeer, 188
Rhinoceros, 150, 173
Roe-deer, 45, 46, 47, 208
Sacculina, 38, 170, 197
Salamander, 17, 203
Sardine, 259
Scallop, 161
Sheep, 17, 88, 89, 90, 153, 188, 201,
222, 223, 238
Solen, 162
Sphaerium, 203
Sphodromantis, 50, 5111., 125, 135
Spider, 125, 261, 262
Spider-crab, 34, 36, 38, 53, 54, 69,
122, 123
Sprat, 116
Stag-beetle, 25, 55, 58, 59, 76, 113,
114, 115, 207, 208, 209
Stick-insect, 27, 175
Stomatopoda, 118
Sun-fish, 104
Swallow-tail, 55, 57, 188
Swan, wild, 225
Tadpole, 173, 182
Telmessus, 93
Tenebrio, 16, 25, 26, 27, 28, 29
Termes, 66 n.
Termites, 65-7, 258
Tern, 263
Theridion, 261
Thrush, 225
Tit, 215
Titanotheria, 218, 219
Tridacna, 162
Trilobites, 134
Triticum, 21
Triton, 179, 180
Tropaeolum, 14, 16, 176 n.
Turnip, 13, 21
Turritella, 161
Uca, 8-12, 31-5, 41, 80, 82, 83, 84, 85.
119, 121, 168, 189, 216, 227, 238
Undulina, 86 n.
Ungulates, 87
Upogebia, 98, 99
Urodele, 135, 181
Vole, 40
Water-boatman, 25, 27
Water-beetle, 25
Wax-moth, 16, 29-31
Whale, 16, 135, 136
Xylotrupes, 61, 72, 76
Zoothamnium, 117
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